Methods and systems for diversity combining of synchronization statistics in OFDM systems

ABSTRACT

Various embodiments of the systems and methods described herein may be used to compute a minimum variance unbiased estimator by receiving a first OFDM signal at a pilot tone, receiving a second OFDM signal sent in the same frequency band and determining a differential phase metric between the first OFDM signal and the second OFDM signal. In some embodiments, the differential phase metric may be used to diversity combine synchronization statistics. In various embodiments, the differential phase metric may be used to detect a narrow-band interference.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of and claims priority to U.S. patentapplication Ser. No. 12/099,017, filed Apr. 7, 2008, which issued asU.S. Pat. No. 8,031,789 and which claimed priority to U.S. ProvisionalPatent Applications: No. 60/910,635, No. 60/910,639, and 60/910,641,each filed Apr. 6, 2007. The contents, each of which, are incorporatedherein by reference in their entirety.

FIELD OF THE INVENTION

The present invention is directed toward methods and systems fordiversity combining of synchronization statistics in OFDM systems.

DESCRIPTION OF THE RELATED ART

With the many continued advancements in communications technology, moreand more devices are being introduced in both the consumer andcommercial sectors with advanced communications capabilities.Additionally, advances in processing power and low-power consumptiontechnologies, as well as advances in data coding techniques have led tothe proliferation of wired and wireless communications capabilities on amore widespread basis.

For example, communication networks, both wired and wireless, are nowcommonplace in many home and office environments. Such networks allowvarious heretofore independent devices to share data and otherinformation to enhance productivity or simply to improve theirconvenience to the user. One such communication network that is gainingwidespread popularity is an exemplary implementation of a wirelessnetwork such as that specified by the WiMedia-MBOA (Multiband OFDMAlliance). Other exemplary networks include the Bluetooth®communications network and various IEEE standards-based networks such as802.11 and 802.16 communications networks, to name a few.

Such networks have proliferated airports, coffee shops, hotels, officesand other public places. Additionally, wireless networking has becomecommonplace in the home environment to provide convenience to the userin easily connecting multiple devices. Architects of these and othernetworks, and indeed communications channels in general, have longstruggled with the challenge of managing multiple communications acrossa limited channel. For example, in some environments, more than onedevice may share a common carrier channel and thus run the risk ofencountering a communication conflict between the one or more devices onthe channel. Over the years, network architects have come up withvarious solutions to arbitrate disputes or otherwise delegate bandwidthamong the various communicating devices, or clients, on the network.Schemes used in well known network configurations such as token rings,Ethernet, and other configurations have been developed to allow sharingof the available bandwidth. In addition to these schemes, othertechniques have been employed, including for example CDMA (code divisionmultiple access) and TDMA (time division multiple access) for cellularnetworks.

FDM (Frequency Division Multiplexing) is another technology that enablesmultiple devices to transmit their signals simultaneously over acommunication channel in a wired or wireless setting. The devices'respective signals travel within their designated frequency band(carrier), onto which the data (text, voice, video, or other data.) ismodulated. With adequate separation in frequency band spacing, multipledevices can simultaneously communicate across the same communicationchannel (network or point-to-point).

Orthogonal FDM (OFDM) spread spectrum systems distribute the data over aplurality of carriers that are spaced apart at precise frequencies. Thespacing is chosen so as to provide orthogonality among the carriers.Thus, a receiver's demodulator recovers the modulated data with littleinterference from the other carrier signals. The benefits of OFDM arehigh spectral efficiency, resiliency to RF interference, and lowermulti-path distortion or inter symbol interference (ISI). OFDM systemscan be combined with other techniques (such as time-divisionmultiplexing) to allow sharing of the individual carriers by multipledevices as well, thus adding another dimension of multiplexingcapability.

With the proliferation of the many different wireless systems,interference from, for example, other transmitters may block a desiredsignal. Additionally, in some cases it can be necessary to estimatecarrier or sampling frequency in various communication systems. Variouscommunication systems might also diversity to improve signal receptionquality.

In various embodiments, the cumulative effect of CFO and SFO might becompensated for in order to limit or avoid the performance loss. Someimplementations treat each OFDM symbol as a separate entity from theperspective of correcting the SFO. These implementations may run a pilotphase tracking algorithm that may be designed to compensate for SFO. Thepilot phase trading algorithm may be run anew for every OFDM symbolthroughout the entire packet.

In some embodiments, synchronization algorithms might not rely onchannel estimation because channel estimators may have inherent errors.This may be especially true if the channel estimators are not toosophisticated. These errors may impair the synchronization algorithms'performance.

In some cases, differential, non-coherent detection may bypass the needfor channel state information. Similarly, good carrier frequency-offsetestimators might rely on some differential metric in order to not dependon the quality of a channel estimator. It was shown in P. H. Moose, “Atechnique for orthogonal frequency division multiplexing frequencyoffset correction,” IEEE Trans. Commun., vol. COMM-42, pp. 2908-2914,October 1994, (“Moose”) that a maximum likelihood (ML) estimate of thecarrier frequency offset (CFO) can be obtained from a differentialmetric. The Moose article is incorporated herein in its entirety byreference. One limitation of the Moose article is that it does notdifferentiate between CFO and sampling (clock) frequency offset (SFO);that distinction was made in M. Speth, S. A. Fechtel, G. Fock, and H.Meyr, “Optimum receiver design for wireless broad-band systems usingOFDM—Part I,” IEEE Trans. Commun., vol. 47, pp. 1668-1677, November1999, (“Speth Part I”), M. Speth, S. A. Fechtel, G. Fock, and H. Meyr,“Optimum receiver design for OFDM-based broad-band transmission—Part II:A case study,” IEEE Trans. Commun., vol. 49, pp. 571-578, April 2001,(“Speth Part II”), which do start from the ML estimate in the Moosearticle, but the CFO and SFO estimators proposed therein lackoptimality. Specifically, they fail to make the connection betweensufficient statistics and ML estimators (which are not necessarilysufficient statistics); they also fail to bridge the qualitative gapbetween sufficient ML estimators (when they exist) and minimum varianceunbiased (MVUB) estimators; what is being lost is the opportunity toachieve the Cramer-Rao bound by construction, due to the simple factthat MVUB estimators based on complete, sufficient statistics are unique(this is the Rao-Blackwell theorem) and, thereby, achieve the Cramer-Raobound. This qualitative refinement is not recognized, discussed orimplemented in prior art. Speth Parts I and II are hereby incorporatedby reference in their entirety.

Further, the derivations of estimators for CFO and SFO in the Speth PartI and II articles rely heavily on Rayleigh fading channel assumptions,which imply that the fading is flat in the frequency domain; this isapparent, to the careful reader, from Section IV.C.2 in Speth Part I andII articles from the applicability and derivation of eq. (14) in SpethPart I, and from the simulation results presented in FIG. 6 of SpethPart II, which is for a Rayleigh channel. While a Rayleigh fading modelis justified in satellite communications channels (negligible excessdelay due to propagation through atmosphere), the same is not the casewith highly frequency-selective channels, such as terrestrial fadingchannels and at least some of the indoors channels that modelpropagation of ultrawide band signals. See J. Foerster, “Channelmodeling sub-committee report final,” Report IEEE P802.15-02/490r1-SG3a,IEEE P802.15 Working Group for Wireless Personal Area Networks, Feb. 7,2003, incorporated herein by reference in its entirety. For thesescenarios, different frequency offset estimation methods are required,which can cope with a channel's strong frequency selectivity duringestimation of the frequency offsets.

Apart from these qualitative distinctions that ultimately do influencetheir performance, estimators based on differential metrics performpilot phase tracking differentially, from one OFDM symbol to thenext—rather than individually for each OFDM symbol, in isolation fromother OFDM symbols; it is this differential feature that correctly anddesirably avoids the estimators' dependence on channel stateinformation. The Speth Part I and II articles are incorporated herein intheir entirety by reference.

From another perspective, solutions known in the art do not leverage theSFO/CFO correction, or compensation, on a form of diversity orredundancy—primarily because wireless OFDM systems are relatively new,and diversity mechanisms that could be relevant in fading channels toCFO/SFO tracking algorithms were lacking; in particular, in OFDM systemseach subcarrier (tone) experiences flat fading, which does not lenditself to diversity mechanisms via signal combining. Moreover, multipathdiversity in the time domain, which could be exploited by signalcombining, translates into frequency selectivity in the frequencydomain, which must be exploited by coding and/or interleaving, ratherthan by signal combining. In addition, the more recent digitalbroadcasting systems based on OFDM, as well as MIMO OFDM systems, do notrely on frequency hopping, which constitutes both a form of spectrumspreading and a source of diversity in fading channels. In essence,neither did SFO/CFO schemes known in the art employ any method forexploiting diversity from frequency hopping, nor did they have clearopportunities to do so.

BRIEF SUMMARY OF THE INVENTION

Various embodiments of the systems and methods described herein may beused to compute a minimum variance unbiased estimator by receiving afirst OFDM signal at a pilot tone, receiving a second OFDM signal sentin the same frequency band and determining a differential phase metricbetween the first OFDM signal and the second OFDM signal.

In some embodiments, the differential phase metric may be used todiversity combine synchronization statistics. For example, in someembodiments a diversity system may determine a plurality of diversityobservations in a diversity system by calculating a plurality ofdifferential phase metrics at a plurality of pilot tones. The diversityobservations may be collected on the same pilot tones during a number ofuncorrelated frequency hops and combine.

In various embodiments, the differential phase metric may be used todetect a narrow-band interference. Various embodiments calculate adifferential phase metric. The metric can then be compared to apredetermined threshold and the likelihood that a narrowband interfereris present can be determined based on the comparison.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention, in accordance with one or more variousembodiments, is described in detail with reference to the followingfigures. The drawings are provided for purposes of illustration only andmerely depict typical or example embodiments of the invention. Thesedrawings are provided to facilitate the reader's understanding of theinvention and shall not be considered limiting of the breadth, scope, orapplicability of the invention. It should be noted that for clarity andease of illustration these drawings are not necessarily made to scale.

FIG. 1 is a block diagram illustrating one possible configuration of awireless network that might serve as an example environment in which thepresent invention can be implemented.

FIG. 2, which comprises FIG. 2A and FIG. 2B, is a flowchart illustratingan example system in accordance with the systems and methods describedherein.

The figures are not intended to be exhaustive or to limit the inventionto the precise form disclosed. It should be understood that theinvention can be practiced with modification and alteration, and thatthe invention be limited only by the claims and the equivalents thereof.

DETAILED DESCRIPTION OF THE EMBODIMENTS OF THE INVENTION

In various embodiments, systems may be implemented to calculate thedifferential phase metric at pilot tone k between two consecutive OFDMsymbols sent in the same frequency band. This translates into observing,in the frequency domain, at least two consecutive OFDM symbols or asmany as the number of active spectral hop bands plus one. In someembodiments, this may be repeated for other pilot tones or all tonesthat are to be used in estimating the carrier and sampling clockrelative frequency offsets. If more than one diversity branch isavailable, the system may collect diversity observations on the samepilot tones. This can be done during a number of uncorrelated diversitybranches, e.g. frequency hops. If more than one diversity branches areavailable some systems may combine diversity observations of the(sufficient) differential synchronization statistic. Various systems canapply the unique minimum variance unbiased estimator for the carrier andsampling clock relative frequency offsets. In some embodiments asymmetric representation of the frequency domain signal, such as aconjugate symmetric component retrieved from the original observablesmay be applied to the unique minimum variance unbiased estimator for thecurrier and sampling clock relative frequency offsets. Without resortingto a symmetric component such as a conjugate symmetric componentretrieved from the original observables the systems and methodsdescribed herein may combine the above estimates, for example, bysumming them. Additionally they may remove the effect of the widebandsignal from the combined estimates based on asymmetric observables.Various embodiments operate on the resulting metric, by comparing it toa selected threshold. This may be performed in order to make thehypotheses that a narrow band interferer is present or absent. In someembodiments, the metric due to a possible narrowband interferer can beprocessed in order to assess its relative strength with respect to thetolerated wideband device and to avoid or mitigate the narrow banddevice operating within the band of the tolerated wideband device. Insome embodiments, a system may either correct the local oscillator(s)frequency, or form and apply a correction to the demodulated samples(post DFT) based on the estimated synchronization parameters. This canbe done in parallel. Additionally, the estimation and detectionprocedure may be repeated throughout the duration of the packet.

Before describing the invention in detail, it is useful to describe anexample environment with which the invention can be implemented. Onesuch example is that of a wireless network. FIG. 1 is a block diagramillustrating one possible configuration of a wireless network that canserve as an example environment in which the present invention can beimplemented. Referring now to FIG. 1, a wireless network 120 is providedto allow a plurality of electronic devices to communicate with oneanother without the need for wires or cables between the devices. Awireless network 120 can vary in coverage area depending on a number offactors or parameters including, for example, the transmit power levelsand receive sensitivities of the various electronic devices associatedwith the network. Examples of wireless networks can include the variousIEEE and other standards as described above, as well as other wirelessnetwork implementations. Another example of a wireless network is thatspecified by the WiMedia standard (within the WiMedia and Multi-BandOFDM Alliance). From time-to-time, the present invention is describedherein in terms of wireless network. Description in terms of theseenvironments is provided to allow the various features and embodimentsof the invention to be portrayed in the context of an exemplaryapplication. After reading this description, it will become apparent toone of ordinary skill in the art how the invention can be implemented indifferent and alternative environments. Indeed, applicability of theinvention is not limited to a wireless network, nor is it limited to aWiMedia standard described as one implementation of the exampleenvironment.

With many applications, the wireless network 120 operates in arelatively confined area, such as, for example, a home or an office. Theexample illustrated in FIG. 1 is an example of an implementation such asthat which may be found in a home or small office environment. Of coursewireless communication networks and communication networks in generalare found in many environments outside the home and office as well. Inthe example illustrated in FIG. 1, wireless network 120 includes acommunication device to allow it to communicate with external networks.More particularly, in the illustrated example, wireless network 120includes a modem 140 to provide connectivity to an external network suchas the Internet 146, and a wireless access point 142 that can provideexternal connectivity to another network 144.

Also illustrated in the example wireless network 120 are portableelectronic devices such as a cellular telephone 110 and a personaldigital assistant (PDA) 112. Like the other electronic devicesillustrated in FIG. 1, cellular telephone 110 and PDA 112 cancommunicate with wireless network 120 via the appropriate wirelessinterface. Additionally, these devices may be configured to furthercommunicate with an external network. For example, cellular telephone110 is typically configured to communicate with a wide area wirelessnetwork by way of a base station.

Additionally, the example environment illustrated in FIG. 1 alsoincludes examples of home entertainment devices connected to wirelessnetwork 120. In the illustrated example, electronic devices such as agaming console 152, a video player 154, a digital camera/camcorder 156,and a high definition television 158 are illustrated as beinginterconnected via wireless network 120. For example, a digital cameraor camcorder 156 can be utilized by a user to capture one or more stillpicture or motion video images. The captured images can be stored in alocal memory or storage device associated with digital camera orcamcorder 156 and ultimately communicated to another electronic devicevia wireless network 120. For example, the user may wish to provide adigital video stream to a high definition television set 158 associatedwith wireless network 120. As another example, the user may wish toupload one or more images from digital camera 156 to his or her personalcomputer 160 or to the Internet 146. This can be accomplished bywireless network 120. Of course, wireless network 120 can be utilized toprovide data, content, and other information sharing on a peer-to-peeror other basis, as the provided examples serve to illustrate.

Also illustrated is a personal computer 160 or other computing deviceconnected to wireless network 120 via a wireless air interface. Asdepicted in the illustrated example, personal computer 160 can alsoprovide connectivity to an external network such as the Internet 146.

In the illustrated example, wireless network 120 is implemented toprovide wireless connectivity to the various electronic devicesassociated therewith. Wireless network 120 allows these devices to sharedata, content, and other information with one another across wirelessnetwork 120. Typically, in such an environment, the electronic deviceswould have the appropriate transmitter, receiver, or transceiver toallow communication via the air interface with other devices associatedwith wireless network 120. These electronic devices may conform to oneor more appropriate wireless standards and, in fact, multiple standardsmay be in play within a given neighborhood. Electronic devicesassociated with the network typically also have control logic or modulesconfigured to manage communications across the network and to manage theoperational functionality of the electronic device. Such control logiccan be implemented using hardware, software, or a combination thereof.For example, one or more processors, ASICs, PLAs, and other logicdevices or components can be included with the device to implement thedesired features and functionality. Additionally, memory or other dataand information storage capacity can be included to facilitate operationof the device and communication across the network.

Electronic devices operating as a part of wireless network 120 aresometimes referred to herein as network devices, members or memberdevices of the network or devices associated with the network. In oneembodiment, devices that communicate with a given network may be membersor merely in communication with the network.

In the WiMedia physical layer standard, as in any OFDM system, it may benecessary to cope with the nonzero sampling frequency offset (SFO). Inone embodiment, this can be done either by (1) estimating it (or aparameter intricately connected to it) then correcting the frequency ofthe local oscillator from which the sampling frequency is derived, or(2) by estimating it and simply compensating for its effect, forexample, correcting the subcarrier phases and/or advancing or retractingthe current sampling instant. The extent to which this estimation isinaccurate may amount to a performance loss. This can be even more sowhen the frequency spacing between sub-carriers is very small. Incombination with the carrier frequency offset (CFO), SFO leads toperformance loss can be due to a rotation of the desired modulatedsymbols. These symbols may be superimposed on the data tones, andforming the object of detection. Performance loss may also be due tointer-carrier interference also called frequency offset noise.

Synchronization in OFDM systems, in particular, estimating the CFO andSFO may be reduced to estimating a differential phase variation(increment) in the complex domain. Two observations in additive whiteGaussian noise (AWGN) are illustration in equation (1).

$\begin{matrix}\left\{ \begin{matrix}{Y_{1} = {R_{1} + v_{1}}} \\{Y_{2} = {{R_{1}{\exp({j\theta})}} + v_{2}}}\end{matrix} \right. & (1)\end{matrix}$In equation (1), all quantities except θ are complex, j=√{square rootover (−1)}, ν₁,ν₂ is complex (circular) AWGN with variance σ_(N) ², andθ is a random phase parameter to be estimated—represents some phaseincrement of the signal component in the new observation Y₂ relative tothe signal component R₁ in the previous observation Y₁. The noise termsverify

$\begin{matrix}{{E\left\{ {v_{1R}v_{1I}} \right\}} = {{E\left\{ {v_{2R}v_{2I}} \right\}} = 0}} & (2) \\{{{var}\left( v_{1} \right)} = {{{var}\left( v_{2} \right)} = \sigma_{N}^{2}}} & (3) \\{{E\left\{ v_{1} \right\}} = {{E\left\{ v_{2} \right\}} = 0}} & (4) \\{{E\left\{ v_{1R}^{2} \right\}} = {{E\left\{ v_{1I}^{2} \right\}} = {{E\left\{ v_{2R}^{2} \right\}} = {{E\left\{ v_{2I}^{2} \right\}} = {\sigma_{N}^{2}/2}}}}} & (5)\end{matrix}$

In OFDM synchronization, the physical meaning of the above model is thatthe observations Y₁,Y₂ represent the complex signal in the frequencydomain at a particular subcarrier. This complex signal may be takenduring two consecutive OFDM symbols. Additionally, the particularsubcarrier may arise as a frequency domain sample of the Fouriertransform of the time domain signal, and the sampling effect isimplemented via the use of the discrete Fourier transform (DFT).

It will be recognized by one of ordinary skill in the art that not onlydoes this model admit a simple, optimal estimator for the relevantCFO/SFO parameters, but the optimal estimator, which is the unique MVUBEthat achieves the Cramer-Rao bound does not rely on, or require, anyinformation about the channel state. Accordingly, any limitations orerrors of the channel estimator implemented in the receiver (for use indata detection and decoding) may be transparent to the optimality of thesynchronization algorithm.

The well-known isomorphism between a complex number and a real vector inthe sequel; e.g., the complex scalar Y₁ is isomorphic to the realelement two dimensional vector Y₁Y ₁ =Y _(1R) +jY _(1I) ˜[Y _(1R) Y _(2I)]^(T) □Y ₁.  (6)

Similarly for Y₂, or other vectors of complex scalars we can let thereal column vector Ξ₂□[Ξ_(2R) Ξ_(2I)]^(T) represent the productY* ₁ Y ₂=(Y _(1R) −jY _(1I))(Y _(2R) +jY _(2I))□Ξ₂□Ξ_(2R) +jΞ_(2I)∈□  (7)via the isomorphism Ξ_(2R)+jΞ_(2I)˜Ξ₂.

The initial complex observations Y₁,Y₂∈□ form a 4-dimensional statistic[Y₁ ^(T)Y₂ ^(T)]^(T). Because the complexity of the implementedestimator for θ depends on the dimensionality of the statistic beingused, one may question whether there exists another statistic of smallerdimensionality that is sufficient for θ. In other words, anotherstatistic that may replace the original one [Y₁ ^(T)Y₂ ^(T)]^(T) with noinformation about θ being lost. Likewise, it is worth knowing how muchcan the sufficient statistic's dimensionality be reduced, i.e. what isthe dimension of the minimal sufficient statistic.

Various embodiments may use a statistic of reduced dimensionality.Consider, for example, the random variables transformation from Y₁,Y₂ toY₁,Ξ₂:

$\begin{matrix}{\begin{bmatrix}Y_{1}^{T} & \Xi_{2}^{T}\end{bmatrix} = {\begin{bmatrix}Y_{1} \\\Xi_{2}\end{bmatrix}^{T} = {{\begin{bmatrix}Y_{1}^{T} & Y_{2}^{T}\end{bmatrix}\begin{bmatrix}I & 0 \\0 & W\end{bmatrix}}\mspace{14mu}\bullet\mspace{20mu}{\overset{\Cap}{w}\left( \begin{bmatrix}Y_{1}^{T} & Y_{2}^{T}\end{bmatrix} \right)}}}} & (8)\end{matrix}$where

$\begin{matrix}{W\mspace{14mu}{\bullet\mspace{14mu}\begin{bmatrix}Y_{1R} & {- Y_{1I}} \\Y_{1I} & Y_{1R}\end{bmatrix}}} & (9)\end{matrix}$

Then

$\begin{matrix}{\begin{bmatrix}Y_{1}^{T} & Y_{2}^{T}\end{bmatrix} = {{\begin{bmatrix}Y_{1}^{T} & \Xi_{2}^{T}\end{bmatrix}\begin{bmatrix}I & 0 \\0 & W^{- 1}\end{bmatrix}}\mspace{14mu}\bullet\mspace{20mu}{{\overset{\Cap}{w}}^{- 1}\left( \begin{bmatrix}Y_{1}^{T} & \Xi_{2}^{T}\end{bmatrix} \right)}}} & (10)\end{matrix}$

The Jacobian of the transformation

⁻¹ isJ([Y ₁ ^(T)Ξ₂ ^(T)])=det(w ⁻¹)=det⁻¹(w)=∥Y ₁∥⁻²=J

−1  (11)and the Jacobian of the transformation

(•) isJ

=(Y _(1R) ² +Y _(1I) ²)=∥Y ₁∥².  (12)

Then, by the transformation of r.v.s from Y₁,Y₂ to Y₁,Ξ₂ one finds thatthe probability density function (pdf) of Y₁,Y₂ conditioned on θ isf(Y ₁ ,Y ₂|θ)=f([Y ₁ ^(T)Ξ₂ ^(T)]=

([Y ₁ ^(T) Y ₂ ^(T)])|θ)|J

|

Then

$\begin{matrix}\begin{matrix}{{f\left( {Y_{1},{Y_{2}❘\theta}} \right)} = {{f\left( {Y_{1},{\Xi_{2}❘\theta}} \right)}{Y_{1}}^{2}}} \\{= {{f\left( {{Y_{1}❘\Xi_{2}},\theta} \right)}{f\left( {\Xi_{2}❘\theta} \right)}{Y_{1}}^{2}}} \\{\overset{(a)}{=}{{f\left( {Y_{1}❘\Xi_{2}} \right)}{f_{\theta}\left( \Xi_{2} \right)}{Y_{1}}^{2}}} \\{= {{f\left( {Y_{1}❘{W^{T}Y_{2}}} \right)}{Y_{1}}^{2}{f_{\theta}\left( \Xi_{2} \right)}}} \\{\bullet{b\left( {Y_{1},Y_{2}} \right)}{f_{\theta}\left( \Xi_{2} \right)}}\end{matrix} & (13)\end{matrix}$where equality (a) follows because Y₁ does not depend on θ, and thenotation f(Ξ₂|θ)=f_(θ)(Ξ₂) is another notation for the pdf of Ξ₂parameterized by the (deterministic) parameter θ. f(Ξ₂|θ) is thedistribution of the random parameter Ξ₂ parameterized by θ orconditioned on θ, depending on whether θ is modeled (viewed) asdeterministic or random. In the latter case, naturally,f(Ξ₂|θ)=f(Ξ₂,θ)/f(θ).

By the Fisher-Neyman factorization theorem, Proposition 1: Ξ₂ is a2-dimensional statistic of the observations [Y₁ ^(T)Y₂ ^(T)]^(T) that issufficient for the parameter θ.

From this preliminary conclusion it can be shown that the ML estimatorderived in the Moose article for θ, based on the differential model (1),can be a function of a sufficient statistic. This will be important whenthe goal becomes not to find an estimate for θ per se, but rather a MVUBestimator for a vector of two parameters (CFO and SFO) that togetherdetermine the phase increment θ in OFDM synchronization problems.

A 2-dimensional complete, sufficient statistic from a 2-parameterexponential family

Now write

$\begin{matrix}\begin{matrix}{{f\left( {Y_{1},{\Xi_{2}❘\theta}} \right)} = {{f\left( {{\Xi_{2}❘Y_{1}},\theta} \right)}{f\left( {Y_{1}❘\theta} \right)}}} \\{= {{f\left( {{\Xi_{2}❘Y_{1}},\theta} \right)}{f\left( Y_{1} \right)}}}\end{matrix} & (14)\end{matrix}$because Y₁ does not depend on θ.

To compute f(Ξ₂|Y₁,θ) note that, conditioned on Y₁, the multiplicationY*₁Y₂ is represented in vector form as

$\begin{matrix}\begin{matrix}{\Xi_{2}^{T} = {Y_{2}^{T}\begin{bmatrix}Y_{1R} & {- Y_{1I}} \\Y_{1I} & Y_{1R}\end{bmatrix}}} \\{= \begin{bmatrix}{{R_{1R}\cos\;\theta} - {R_{1I}\sin\;\theta} + v_{2R}} & {{R_{1R}\sin\;\theta} - {R_{1I}\cos\;\theta} + v_{2I}}\end{bmatrix}} \\{\begin{bmatrix}Y_{1R} & {- Y_{1I}} \\Y_{1I} & Y_{1R}\end{bmatrix}} \\{= \begin{bmatrix}\begin{matrix}{{R_{1R}Y_{1R}\cos\;\theta} - {R_{1I}Y_{1R}\sin\;\theta} + {v_{2R}Y_{1R}} +} \\{{R_{1R}Y_{1I}\sin\;\theta} + {R_{1I}Y_{1I}\cos\;\theta} + {v_{2I}Y_{1I}}}\end{matrix} \\\begin{matrix}{{{- R_{1R}}Y_{1I}\cos\;\theta} - {R_{1I}Y_{1I}\sin\;\theta} - {v_{2R}Y_{1I}} +} \\{{R_{1R}Y_{1R}\sin\;\theta} + {R_{1I}Y_{1R}\cos\;\theta} + {v_{2I}Y_{1R}}}\end{matrix}\end{bmatrix}^{T}} \\{= {\begin{bmatrix}{\xi_{2R} + {\left( {v_{2}Y_{1}^{*}} \right)}} \\{\xi_{2I} + {\left( {v_{2}Y_{1}^{*}} \right)}}\end{bmatrix}^{T} = \begin{bmatrix}{\xi_{2R} + n_{2R}} \\{\xi_{2I} + n_{2I}}\end{bmatrix}^{T}}}\end{matrix} & (15)\end{matrix}$wheren ₂□ν₂ Y* ₁  (16)and, conditioned on Y₁,

$\begin{matrix}{\mspace{79mu}{{E\left\{ n_{2} \right\}} = 0}} & (17) \\\begin{matrix}{{E\left\{ {n_{2R}n_{2I}} \right\}} = {E\left\{ {\left( {{v_{2R}Y_{1R}} + {v_{2I}Y_{1I}}} \right)\left( {{v_{2I}Y_{1R}} - {v_{2R}Y_{1I}}} \right)} \right\}}} \\{= {{E\left\{ {{v_{2R}v_{2I}Y_{1R}^{2}} - {v_{2R}^{2}Y_{1R}Y_{1I}} + {v_{2I}^{2}Y_{1R}Y_{1I}} - {v_{2I}v_{2R}Y_{1I}^{2}}} \right\}} = 0}}\end{matrix} & (18) \\{\mspace{79mu}{{{var}\left( n_{2} \right)} = {{Y_{1}}^{2}\mspace{14mu}{{var}\left( v_{2} \right)}}}} & (19) \\{\mspace{79mu}{{{var}\left( n_{2R} \right)} = {{{var}\left( n_{2I} \right)} = {{Y_{1}}^{2}{{{var}\left( v_{2} \right)}/2}}}}} & (20)\end{matrix}$

ButY ₁ Y* ₁ =|Y ₁|² =R ₁ Y* ₁+ν₁ Y* ₁=(R _(1R) Y _(1R) +R _(1I) Y_(1I))+j(R _(1I) Y _(1R) −R _(1R) Y _(1I))+ν₁ Y* ₁  (21)and

$\begin{matrix}\begin{matrix}{\xi_{2R} = {{\left( {{R_{1R}Y_{1R}} + {R_{1I}Y_{1I}}} \right)\cos\;\theta} + {\left( {{R_{1I}Y_{1R}} - {R_{1R}Y_{1I}}} \right)\sin\;\theta}}} \\{= {{\left( {{Y_{1}}^{2} - {\left( {v_{1}Y_{1}^{*}} \right)}} \right)\cos\;\theta} + {\left( {v_{1}Y_{1}^{*}} \right)\sin\;\theta}}} \\{= {{\left( {{Y_{1}}^{2} - n_{1R}} \right)\cos\;\theta} + {n_{1I}\sin\;\theta}}}\end{matrix} & (22)\end{matrix}$wheren ₁□ν₁ Y* ₁  (23)and, conditioned on Y₁,

$\begin{matrix}{{E\left\{ n_{1} \right\}} = 0} & (24) \\\begin{matrix}{{E\left\{ {n_{1R}n_{1I}} \right\}} = {E\left\{ {\left( {{v_{1R}Y_{1R}} + {v_{1I}Y_{1I}}} \right)\left( {{v_{1I}Y_{1R}} + {v_{1R}Y_{1I}}} \right)} \right\}}} \\{= {E\left\{ {{v_{1R}v_{1I}Y_{1R}^{2}} - {v_{1R}^{2}Y_{1R}Y_{1I}} + {v_{1I}^{2}Y_{1R}Y_{1I}} -} \right.}} \\{\left. {v_{1I}v_{1R}Y_{1I}^{2}} \right\} = 0}\end{matrix} & (25) \\{{{var}\left( n_{1} \right)} = {{Y_{1}}^{2}{{var}\left( v_{1} \right)}}} & (26) \\{{{var}\left( n_{1R} \right)} = {{{var}\left( n_{1I} \right)} = {{{Y_{1}}^{2}{{{var}\left( v_{1} \right)}/2}} = {{{var}\left( n_{2R} \right)} = {{var}\left( n_{2I} \right)}}}}} & (27) \\{{E\left\{ {n_{1}n_{2}^{*}} \right\}} = {{{Y_{1}}^{2}E\left\{ {v_{1}v_{2}^{*}} \right\}} = 0}} & (28)\end{matrix}$

Thenξ_(2R) =|Y ₁ ² cos θ−(n _(1R) cos θ−n _(1I) sin θ)  (29)and similarly

$\begin{matrix}\begin{matrix}{\xi_{21} = {{\left( {{R_{1I}Y_{1R}} - {R_{1R}Y_{1I}}} \right)\cos\;\theta} + {\left( {{R_{1I}Y_{1I}} + {R_{1R}Y_{1R}}} \right)\sin\;\theta}}} \\{= -} \\{= {{{Y_{1}}^{2}\sin\;\theta} - {\left( {{n_{1I}\cos\;\theta} + {n_{1R}\sin\;\theta}} \right).}}}\end{matrix} & (30)\end{matrix}$

Thereby

$\begin{matrix}{\Xi_{2}^{T} = {\begin{bmatrix}\Xi_{2R} & \Xi_{2I}\end{bmatrix} = {\begin{bmatrix}{{{Y_{1}}^{2}\cos\;\theta} - {n_{1R}\cos\;\theta} + {n_{1I}\sin\;\theta} + n_{2R}} \\{{{Y_{1}}^{2}\sin\;\theta} - {n_{1I}\cos\;\theta} + {n_{1R}\sin\;\theta} + n_{2I}}\end{bmatrix}^{T} = {\quad{\begin{bmatrix}{{{Y_{1}}^{2}\cos\;\theta} + \upsilon_{2R}} & {{{Y_{1}}^{2}\sin\;\theta} + \upsilon_{2I}}\end{bmatrix}.}}}}} & (31)\end{matrix}$

Also, it can be verified by straightforward calculations that

$\begin{matrix}{{E\left\{ {\upsilon_{2R}\upsilon_{2I}} \right\}} = {{E\left\{ {\left( {{{- n_{1R}}\cos\;\theta} + {n_{1I}\sin\;\theta} + n_{2R}} \right)\left( {{{- n_{1I}}\cos\;\theta} - {n_{1R}\sin\;\theta} + n_{2I}} \right)} \right\}} = {{E\left\{ {{n_{1R}n_{1I}\cos^{2}\theta} + {n_{1R}^{2}\cos\;{\theta sin}\;\theta} - {n_{1R}n_{2I}\cos\;\theta} - {n_{1I}^{2}\sin\;{\theta cos\theta}} - {n_{1I}n_{1R}\sin^{2}\theta} + {n_{1I}n_{2I}\sin\;\theta} - \;{n_{1I}n_{2R}\cos\;\theta} - {n_{1R}n_{2R}\sin\;\theta} + {n_{2R}n_{2I}}} \right\}} = 0}}} & (32)\end{matrix}$

Because ν_(2R), ν_(2I) are uncorrelated and Gaussian, they areindependent, too, conditioned on Y₁. Further,

$\begin{matrix}\begin{matrix}{{{var}\left( \upsilon_{2R} \right)} = {{{{var}\left( n_{1I} \right)}\sin^{2}\theta} + {{{var}\left( n_{1I} \right)}\cos^{2}\theta} + {{var}\left( n_{2R} \right)}}} \\{= {{{{Y_{1}}^{2}{{{var}\left( v_{1} \right)}/2}} + {{Y_{1}}^{2}{{{var}\left( v_{2} \right)}/2}}} = {{Y_{1}}^{2}{{var}\left( v_{1} \right)}}}}\end{matrix} & (33) \\{{{var}\left( \upsilon_{2I} \right)} = {{{var}\left( \upsilon_{2R} \right)} = {{Y_{1}}^{2}{{var}\left( v_{1} \right)}}}} & (34) \\{{{var}\left( \upsilon_{2} \right)} = {{2{{var}\left( \upsilon_{2R} \right)}} = {2{Y_{1}}^{2}{{var}\left( v_{1} \right)}}}} & (35)\end{matrix}$

Thereby,

$\begin{matrix}{{{{f\left( {{\Xi_{2}❘Y_{1}},\theta} \right)} =}\quad}{\quad{{{\quad\quad}{\quad\quad}\frac{1}{\sqrt{2\pi}\sigma_{\upsilon_{2R}}}{\exp\left( {- \frac{\left( {\Xi_{2R} - {{Y_{1}}^{2}\cos\;\theta}} \right)^{2}}{2\sigma_{\upsilon_{2R}}^{2}}} \right)}\frac{1}{\sqrt{2\pi}\sigma_{\upsilon_{2R}}}{\exp\left( {- \frac{\left( {\Xi_{2I} - {{Y_{1}}^{2}\sin\;\theta}} \right)^{2}}{2\sigma_{\upsilon_{2I}}^{2}}} \right)}} = {{\frac{1}{2{\pi\sigma}_{\upsilon_{2R}}\sigma_{\upsilon_{2I}}}{\exp\left( {- \frac{\Xi_{2R}^{2} + \Xi_{2I}^{2}}{\sigma_{\upsilon_{2}}^{2}}} \right)}{{\exp\left( {- \frac{{{Y_{1}}^{4}\cos^{2}\theta} + {{Y_{1}}^{4}\sin^{2}\theta}}{\sigma_{\upsilon_{2}}^{2}}} \right)} \cdot {\exp\left( \frac{{2\Xi_{2R}{Y_{1}}^{2}\cos\;\theta} + {2\Xi_{2I}{Y_{1}}^{2}\sin\;\theta}}{\sigma_{\upsilon_{2}}^{2}} \right)}}} = {\frac{1}{{\pi\sigma}_{\upsilon_{2}}^{2}}{\exp\left( {- \frac{{\Xi_{2}}^{2}}{\sigma_{\upsilon_{2}}^{2}}} \right)}{{\exp\left( {- \frac{{Y_{1}}^{4}}{\sigma_{\upsilon_{2}}^{2}}} \right)} \cdot {\exp\left( {\frac{2\Xi_{2R}{Y_{1}}^{2}\cos\;\theta}{\sigma_{\upsilon_{2}}^{2}} + \frac{2\Xi_{2I}{Y_{1}}^{2}\sin\;\theta}{\sigma_{\upsilon_{2}}^{2}}} \right)}}}}}}} & (36)\end{matrix}$

Because ∥Ξ₂∥²=∥Y₂∥²∥Y₁∥²,

$\begin{matrix}{{f\left( {{\Xi_{2}❘Y_{1}},\theta} \right)} = {\frac{1}{{\pi 2}{Y_{1}}^{2}\sigma_{v_{1}}^{2}}{\exp\left( {- \frac{{Y_{2}}^{2}}{2\sigma_{v_{1}}^{2}}} \right)}{{\exp\left( {- \frac{{Y_{1}}^{2}}{2\sigma_{v_{1}}^{2}}} \right)} \cdot {\exp\left( {\frac{\Xi_{2R}\cos\;\theta}{\sigma_{v_{1}}^{2}} + \frac{\Xi_{2I}\sin\;\theta}{\sigma_{v_{1}}^{2}}} \right)}}}} & (37)\end{matrix}$

Then, from (13) and (14)

$\begin{matrix}\begin{matrix}{{f\left( {Y_{1},{Y_{2}❘\theta}} \right)} = {{f\left( {{\Xi_{2}❘Y_{1}},\theta} \right)}{f\left( Y_{1} \right)}{Y_{1}}^{2}}} \\{= {\frac{1}{{\pi 2}{Y_{1}}^{2}\sigma_{v_{1}}^{2}}{{\exp\left( {- \frac{{Y_{2}}^{2} + {Y_{1}}^{2}}{2\sigma_{v_{1}}^{2}}} \right)} \cdot}}} \\{{\exp\left( \frac{{\cos\;{\theta\Xi}_{2R}} + {\sin\;{\theta\Xi}_{2I}}}{\sigma_{v_{1}}^{2}} \right)}{f\left( Y_{1} \right)}{Y_{1}}^{2}} \\{= {\frac{1}{2{\pi\sigma}_{v_{1}}^{2}}{{\exp\left( {- \frac{{Y_{2}}^{2} + {Y_{1}}^{2}}{2\sigma_{v_{1}}^{2}}} \right)} \cdot}}} \\{{\exp\left( \frac{{\Xi_{2R}\cos\;\theta} + {\Xi_{2I}\sin\;\theta}}{\sigma_{v_{1}}^{2}} \right)}{{f\left( Y_{1} \right)}.}}\end{matrix} & (38)\end{matrix}$

Denoting by x the initial 4-dimensional statistic x□[Y₁ ^(T)Y₂^(T)]^(T), f_(θ)([Y₁ ^(T)Y₂ ^(T)]^(T)) has the form

${c(\theta)}{a(x)}{\exp\left( \left\{ {\sum\limits_{i = 1}^{2}\;{{\pi_{i}(\theta)}{t_{i}(x)}}} \right\} \right)}$with π₁(θ)=cos θ, π₂(θ)=sin θ, t₁(x)=Ξ_(2R), t₂(x)=Ξ_(2I); f(Y₁,Y₂|θ)belongs thereby to a 2-parameter exponential family and t(x)=[t₁(x)t₂(x)]^(T)=[Ξ_(2R) Ξ_(2I)]^(T) is a 2-dimensional sufficient statisticfor θ that is also complete, and thereby minimal, by virtue of theproperties of exponential distribution families. This, along with (32),(31) is summarized in Proposition 2: Ξ₂, the 2-dimensional statistic ofthe observations [Y₁ ^(T)Y₂ ^(T)]^(T), is a complete, and therebyminimal, sufficient statistic for the parameter θ of (1); the twocomponents of the sufficient statistic are affected by independentadditive Gaussian noise terms.

Various embodiments can use an alternative expression for the minimalsufficient statistic. For example, an alternative perspective on theexpression of the sufficient statistic [Ξ_(2R) Ξ_(2I)]^(T) is obtainedvia

$\begin{matrix}\begin{matrix}{{\Xi_{2R} + {j\;\Xi_{2I}}} = {{Y_{2}Y_{1}^{*}} = {\left( {{R_{1}{\exp\left( {j\;\theta} \right)}} + v_{2}} \right)\left( {R_{1}^{*} + v_{1}^{*}} \right)}}} \\{= {{{R_{1}}^{2}{\exp\left( {j\;\theta} \right)}} + {v_{2}R_{1}^{*}} + {R_{1}{\exp\left( {j\;\theta} \right)}v_{1}^{*}} + {v_{2}v_{1}^{*}}}} \\{= {{{R_{1}}^{2}\cos\;\theta} + {j{R_{1}}^{2}\sin\;\theta} + {v_{2R}R_{1R}} +}} \\{{v_{2I}R_{1I}}\; + {j\left( {{{- v_{2R}}R_{1I}} + {v_{2I}R_{1R}}} \right)} +} \\{\left\lbrack {\left( {{R_{1R}\cos\;\theta} - {R_{1I}\sin\;\theta}} \right) + {j\left( {{R_{1R}\sin\;\theta} +} \right.}} \right.} \\{{\left. {R_{1I}\cos\;\theta} \right\rbrack\left( {v_{1R} - {j\; v_{1I}}} \right)} +} \\{\left( {v_{2R} + {j\; v_{2I}}} \right)\left( {v_{1R} - {j\; v_{1I}}} \right)} \\{= {{{R_{1}}^{2}\cos\;\theta} + {j{R_{1}}^{2}\sin\;\theta} + {v_{2R}R_{1R}} +}} \\{{v_{2I}R_{1I}} + {j\left( {{{- v_{2R}}R_{1I}} + {v_{2I}R_{1R}}} \right)} +} \\{\left\lbrack {{\left( {{R_{1R}\cos\;\theta} - {R_{1I}\sin\;\theta}} \right)v_{1R}} + \left( {{R_{1R}\sin\;\theta} +} \right.} \right.} \\{\left. {\left. {R_{1I}\cos\;\theta} \right)v_{1I}} \right\rbrack + {j\left\lbrack \left( {{R_{1R}\sin\;\theta} +} \right. \right.}} \\{{\left. {R_{1I}\cos\;\theta} \right)v_{1R}} - \left( {{R_{1R}\cos\;\theta} -} \right.} \\{\left. {\left. {R_{1I}\sin\;\theta} \right)v_{1I}} \right\rbrack + \left( {{v_{2R}v_{1R}} + {v_{2I}v_{1I}}} \right) +} \\{j\left( {{v_{2I}v_{1R}} - {v_{2R}v_{1I}}} \right)}\end{matrix} & (39) \\{\Xi_{2R} = {{{{R_{1}}^{2}\cos\;\theta} + {v_{2R}R_{1R}} + {v_{2I}R_{1I}} + \left\lbrack {{\left( {{R_{1R}\cos\;\theta} - {R_{1I}\sin\;\theta}} \right)v_{1R}} + {\left( {{R_{1R}\sin\;\theta} + {R_{1I}\cos\;\theta}} \right)v_{1I}}} \right\rbrack + \underset{\underset{\mu_{2R}^{''}}{︸}}{{v_{2R}v_{1R}} + {v_{2I}v_{1I}}}} = {{{R_{1}}^{2}\cos\;\theta} + \mu_{2R}^{\prime} + {\mu_{2R}^{''}\bullet{R_{1}}^{2}\cos\;\theta} + \mu_{2R}}}} & (40) \\{\Xi_{2I} = {{{{R_{1}}^{2}\sin\;\theta} + \left( {{{- v_{2R}}R_{1I}} + {v_{2I}R_{1R}}} \right) + \left\lbrack {{\left( {{R_{1R}\sin\;\theta} + {R_{1I}\cos\;\theta}} \right)v_{1R}} - {\left( {{R_{1R}\cos\;\theta} - {R_{1I}\sin\;\theta}} \right)v_{1I}}} \right\rbrack + \underset{\underset{\mu_{2I}^{''}}{︸}}{{v_{2I}v_{1R}} - {v_{2R}v_{1I}}}} = {{{R_{1}}^{2}\sin\;\theta} + \mu_{2I}^{\prime} + \mu_{2I}^{''}}}} & (41)\end{matrix}$

The correlation between μ_(2R) and μ_(2I) may be determined as:

$\begin{matrix}\begin{matrix}{{E\left\{ {\mu_{2R}\mu_{2I}} \right\}} = {E\left\{ {\left( {\mu_{2R}^{\prime} + \mu_{2R}^{''}} \right)\left( {\mu_{2I}^{\prime} + \mu_{2I}^{''}} \right)} \right\}}} \\{= {{E\left\{ {\mu_{2R}^{\prime}\mu_{2I}^{\prime}} \right\}} + {E\left\{ {\mu_{2R}^{\prime}\mu_{2I}^{''}} \right\}} +}} \\{{E\left\{ {\mu_{2R}^{''}\mu_{2I}^{\prime}} \right\}} + {E\left\{ {\mu_{2R}^{''}\mu_{2I}^{''}} \right\}}}\end{matrix} & (42)\end{matrix}$

It can be verified by straightforward calculations, albeit cumbersomeand lengthy, that each of the terms in (42) vanish andE{μ _(2R)μ_(2I)}=0  (43)

Therefore, the additive noise terms in the two components of thesufficient statistic for θ are uncorrelated. However, neither μ_(2R) norμ_(2I) are strictly speaking Gaussian, and without normality (43) doesnot imply independence of μ_(2R) from μ_(2I). Nonetheless, theexpressions involving Ξ_(2R) and Ξ_(2I) in the actual estimator(s) willenable the use of the central limit theorem, and will thereby be seen toallow one to separate the averaging operations over expressionscontaining μ_(2R) and respectively μ_(2I).

One embodiment may include an application to CFO and SFO estimation. Thesufficient statistic [Ξ_(2R) Ξ_(2I)]^(T) may be used directly to obtaina ML estimate of the phase parameter θ as in the Moose article, via{circumflex over (θ)}□ arctan(Ξ_(2I)/Ξ_(2R)), where {circumflex over(•)} denotes an estimate.

This ML estimate may have several drawbacks. In some embodiments, it mayonly estimate the differential phase θ. Additionally, it might notreadily distinguish between components of θ that depend on othermeaningful parameters, such as the carrier and sampling clock frequencyoffsets relevant to OFDM synchronization. In addition, maximization ofthe likelihood probability may not minimize the estimator's variance,nor may it guarantee the estimator's performance near the Cramer-Raobound; among ML estimators some perform better than others in term ofestimation error variance. Besides, this estimator is generally notunbiased, as can be seen from Jensen's inequality, and the fact that thetangent function is not linear; depending on the size ofarg(Ξ_(2R)+jΞ_(2I)) the expectation of the estimate, E{{circumflex over(θ)}}, may deviate substantially from θ.

The OFDM synchronization problem seeks to prevent degradation of the SNRseen by the outer receiver due to carrier frequency offset (CFO) and thesampling clock frequency offset (SFO); in OFDM timing andsynchronization issues are corrected both before and after DFT. Thesynchronization problem relevant herein deals with observations in thefrequency domain, i.e. post DFT. Specifically of interest will be thepost DFT fractional carrier frequency offset and the relative samplingclock frequency offset.

It can be shown that the demodulated signal in the frequency domain,during the l-th OFDM symbol and at the k-th subcarrier, post DFT, andwith the appropriate CFO and SFO parameters incorporated in the model,isz _(l,k)=exp(jπΦ _(k))exp[j2π((lN _(s) +N _(g))/N)Φ_(k)]α(Φ_(k))a _(l,k)H _(k) +n _(Ω;l,k) +n _(l,k)  (44)where N_(s)□N+N_(g), N represents the size of the DFT transform, N_(g)is the number of samples contained in the cyclic prefix (or zero paddinginterval), N_(s) represents the number of samples in the extended OFDMsymbol, a_(l,k) is the complex symbol on the k-th subcarrier of the l-thOFDM symbol, H_(k) is the frequency domain channel coefficientassociated with the k-th subcarrier, which is distributed as acircularly symmetric Gaussian random variable.

The AWGN noise term n_(l,k) models in the frequency domain the AWGN inthe time domain; n_(Ω;l,k) is the ICI, or frequency offset noise. Thefactor α(Φ_(k)) is a notation for a ratio of sine functions,sin(πΦ_(k))/sin(πΦ_(k)/N), which has value N when Φ_(k)=0, and can beapproximated by N when Φ_(k) is very small. When the synchronizationactually controls the local oscillators that supply the sampling clockand carrier frequencies, the approximation α(Φ_(k))≈N is very good, andthe fact that the phase error Φ_(k)≠0 is not quite 0 is practicallyinconsequential. However, in the case when the Fourier transform size islarge, and attempts are not made to control the local oscillators'frequencies—but, rather, the receiver algorithms attempt to simplyestimate these offsets and compensate for them—the phase error Φ_(k) canbe large enough, for large k, to cause α(Φ_(k)) to depart from itsdesired maximum, N, at the origin. In these cases the CFO and SFO canlead to significant SNR loss.

The differential phase increment can be modeled by the phase Φ_(k),which is given by:Φ_(k) =ΔfT _(u) +ζk  (45)where Δf is the carrier frequency offset, T_(u) is the duration of theproper OFDM symbol (cyclic prefix excluded, i.e. T_(u) ⁻¹ is thesubcarrier spacing), and ΔfT_(u) is the carrier frequency offsetnormalized to the subcarrier spacing. ζ□(T′−T)/T=(1/T−1/T′)/1/T′ is thesampling frequency offset normalized to the receiver's sampling clockfrequency. The normalized CFO ΔfT_(u) is Δf′□ΔfT_(u)=n₁+Δf′_(F), withΔf′_(F)<1. The integer normalized CFO is manifested in a shift of thefrequency domain DFT samples, and is corrected separately from thefractional normalized carrier frequency offset Δf′_(F).

In the sequel denoten _(Ω;l,k) +n _(l,k)□ν_(l,k).  (46)The factor exp(jπΦ_(k)) cannot be distinguished from the channel gainH_(k), and may thus be (viewed as) incorporated in H_(k).

The approximations discussed above are largely irrelevant to the problemof estimating the carrier and sampling frequency offsets—because thesufficient statistic is based on a differential metric that only needsto model the phase increment induced by imperfect synchronization fromone OFDM symbol to the next. This means that synchronization may remaintransparent to knowledge of CSI (via H_(k)). Additionally, becauserelying on channel estimates would impair synchronization due toinherent channel estimation errors it may be advantageous to synchronizewithout using such estimator.

The following analysis assumes that either the integer normalized CFOhas been corrected post DFT and only the fractional normalized carrierfrequency offset Δf′_(F) remains to be corrected or that the initial CFOΔf is less than the subcarrier spacing T_(u) ⁻¹. The latter can be thecase, for example, when the Fourier transform size N is small. When theFourier transform size is small, the intercarrier spacing is larger thantwice the tolerance of the frequency reference used by the receiver'slocal oscillator. Accordingly, the handling of the two components can beseparated from each other.

Consider the problem formulated in terms of the above set-up as

$\begin{matrix}\begin{matrix}{z_{{l - 1},k} = {R_{{l - 1},k} + v_{{l - 1},k}}} \\{z_{l,k} = {{R_{{l - 1},k}{\exp\left( {j\Delta\varphi}_{k,{(1)}} \right)}} + v_{l,k}}}\end{matrix} & (47)\end{matrix}$where Δφ_(k,(1)) is the variation of the k-th subcarrier's phase due tononzero CFO and SFO, according to (44), (45), over the duration of oneOFDM symbol, i.e. from one OFDM symbol to the next; also,|R _(l−1,k)|²=α²(Φ_(k))|a _(l,k)|² |H _(k)|²;  (48)if the synchronization limits itself to subcarriers that carry pilotsymbols of energy β²σ_(a) ², with σ_(a) ² the energy of the complexconstellation used for pilots and β²>1 the energy-boost factor for thepilot symbols, then|R _(l−1,k)|²=α²(φ_(k))β²σ_(a) ² |H _(k)|².  (49)

This is the same model as the one in (1), used to derive the sufficientstatistic; according to above results, a sufficient, complete andminimal statistic for ΔΦ_(k(1)) isx _(l,k) □z _(l,k) z* _(l−1,k).  (50)

The only assumption made is the benign assumption that the channel doesnot change for the duration necessary to compute the differentialmetric. Let C be the set of indices of the subcarriers that are to beused for CFO and SFO estimation; typically these symbols are pilotsymbols, sometimes implemented as continuous or scattered pilots. C isassumed to be symmetric around 0, i.e. if k∈C then −k∈C; it is useful todenote the subset of positive indices in C as C₊.

Considering, as discussed above, only the fractional (normalized)carrier frequency offset Δf_(F)T_(u), the phase variation on subcarriers−k and k, according to (44), (45), over the duration of one OFDM symbol,i.e. from one OFDM symbol to the next (index l incremented only by 1),is

$\begin{matrix}\begin{matrix}{{\Delta\varphi}_{{\pm k},{(1)}} = {{2{\pi\left( {\left( {N + N_{g}} \right)/N} \right)}\phi_{k}} =}} \\{2{\pi\left( {\left( {N + N_{g}} \right)/N} \right)}\left( {{\Delta\; f_{F}T_{u}} \pm {\zeta\; k}} \right)} \\{= {{2{\pi\left( {\left( {N + N_{g}} \right)/N} \right)}\Delta\; f_{F}T_{u}} \pm {2{\pi\left( {\left( {N + N_{g}} \right)/N} \right)}\zeta\; k}}} \\{{\bullet\Delta\; f_{F}^{''}} \pm {\zeta^{\prime}{k.}}}\end{matrix} & (51)\end{matrix}$

Similarly, the phase variation on subcarriers −k and k, according to(44), (45), over the duration of γ consecutive OFDM symbols is

$\begin{matrix}\begin{matrix}{{\Delta\varphi}_{{\pm k},{(\gamma)}} = {{2{{\pi\gamma}\left( {\left( {N + N_{g}} \right)/N} \right)}\phi_{k}} =}} \\{2{{\pi\gamma}\left( {\left( {N + N_{g}} \right)/N} \right)}\left( {{\Delta\; f_{F}T_{u}} \pm {\zeta\; k}} \right)} \\{= {{{\gamma\Delta}\; f_{F}^{''}} \pm {{\gamma\zeta}^{\prime}{k.}}}}\end{matrix} & (52)\end{matrix}$

Considering the result in discussed above,

x_(l,k),

x_(l,k),

x_(l,−k),

x_(l,−k) form a complete, minimal, sufficient statistic for theparameter vectora _(k)□[Δφ_(k,(1)) Δφ_(−k,(1))]^(T) , k>0,  (53)

In one embodiment conjugate-symmetry reduction of frequency domainobservations may be used. It will be useful in some cases to retrieveeither the conjugate symmetric or the conjugate anti-symmetric componentof the DFT sequences z_(l−1,k) and z_(l,k), which in general do notexhibit conjugate symmetry. The former is accomplished by the simpleoperation

$\begin{matrix}\begin{matrix}{z_{{{CS};{l - 1}},k}\bullet\frac{1}{2}\left( {z_{{l - 1},k} + z_{{l - 1},{- k}}^{*}} \right)} \\{= {{\frac{1}{2}\left( {R_{{l - 1},k} + R_{{l - 1},{- k}}^{*}} \right)} + {\frac{1}{2}\left( {v_{{l - 1},k} + v_{{l - 1},{- k}}^{*}} \right)}}} \\{= {R_{{{CS};{l - 1}},k} + v_{{{CS};{l - 1}},k}}}\end{matrix} & (54)\end{matrix}$where z_(CS;l−1,l), R_(CS;l−1,k), and ν_(CS;l−1,k) are the conjugatesymmetric components of the sequences

{z_(CS; l − 1, k)}_(k = 0)^(N − 1),etc. Similarly, using (51),

$\begin{matrix}\begin{matrix}{z_{{{CS};l},k}\bullet\frac{1}{2}\left( {z_{l,k} + z_{l,{- k}}^{*}} \right)} \\{= {{\frac{1}{2}\left( {{R_{{l - 1},k}{\exp\left( {j\;{\Delta\varphi}_{k,{(1)}}} \right)}} + {R_{{l - 1},{- k}}^{*}{\exp\left( {{- j}\;{\Delta\varphi}_{{- k},{(1)}}} \right)}}} \right)} +}} \\{\frac{1}{2}\left( {v_{l,k} + v_{l,{- k}}^{*}} \right)} \\{= {\frac{1}{2}\left( {{R_{{l - 1},k}{\exp\left( {j\;\Delta\; f_{F}^{''}} \right)}{\exp\left( {j\;\zeta^{\prime}k} \right)}} +} \right.}} \\{\left. {R_{{l - 1},{- k}}^{*}{\exp\left( {{- j}\;\Delta\; f_{F}^{''}} \right)}{\exp\left( {- {j\left( {{- \zeta^{\prime}}k} \right)}} \right)}} \right) + v_{{{CS};l},k}} \\{= {\frac{1}{2}\left( {{R_{{l - 1},k}{\exp\left( {j\;\Delta\; f_{F}^{''}} \right)}} +} \right.}} \\{{\left. {R_{{l - 1},{- k}}^{*}{\exp\left( {{- j}\;\Delta\; f_{F}^{''}} \right)}} \right){\exp\left( {j\;\zeta^{\prime}k} \right)}} + v_{{{CS};l},k}} \\{= {{\frac{1}{2}\left( {R_{{l - 1},k} + {R_{{l - 1},{- k}}^{*}{\exp\left( {{- j}\; 2\Delta\; f_{F}^{''}} \right)}}} \right){\exp\left( {j\;\Delta\; f_{F}^{''}} \right)}{\exp\left( {j\;\zeta^{\prime}k} \right)}} +}} \\{v_{{{CS};l},k}} \\{= {{\frac{1}{2}\left( {R_{{l - 1},k} + {R_{{l - 1},{- k}}^{*}{\exp\left( {{- j}\; 2\Delta\; f_{F}^{''}} \right)}}} \right){\exp\left( {j\;{\Delta\varphi}_{k,{(1)}}} \right)}} + v_{{{CS};l},k}}} \\{\approx {{\frac{1}{2}\left( {R_{{l - 1},k} + R_{{l - 1},{- k}}^{*}} \right){\exp\left( {j\;{\Delta\varphi}_{k,{(1)}}} \right)}} + v_{{{CS};l},k}}} \\{= {{R_{{{CS};{l - 1}},k}{\exp\left( {j\;{\Delta\varphi}_{k,{(1)}}} \right)}} + v_{{{CS};l},k}}}\end{matrix} & (55)\end{matrix}$where the approximation R*_(l−1,−k) exp(−j2Δf″_(F))≈R*_(l−1,−k) isjustified whenever the fractional (or even total) CFO Δf″_(F) is muchsmaller than the phases of at least one of the factors of R*_(l−1,−k).This is de facto the case in all practical implementations; e.g.,R_(l−1,k)=α(Φ_(k))a_(l,k)H_(k), where a_(l,k) is a complex symbol, be itpilot or data, from a (possibly finite size) constellation, such asM-QAM; e.g., in the case of 4-QAM, the phase of a_(l,k) is a multiple ofπ/4□Δf″_(F) whereΔf″_(F)=2π((N+N_(g))/N)Δf_(F)T_(u)=2π((N+N_(g))/N)Δf′_(F) and Δf′_(F) isno larger than 100 ppm (parts per million).

Equations (54), (55) have the same form as (47), and thereby the resultsapply to z_(CS;l−1,k), z_(CS;l,k) as they did to z_(l−1,k), z_(l,k),with the distinction that z_(CS;l−1,k), z_(CS;l,k) have conjugatesymmetry and verify |z_(CS;l,k)|=|z_(CS;l,−k)|; note that this operationreduces the noise variance, i.e. ν_(CS;l,k) has half the variance ofν_(l,k).

When the conjugate symmetry is necessary, z_(CS;l−1,k), z_(CS;l,k) canbe used in place of z_(l−1,k), z_(l,k). A similar discussion applies tousing the conjugate anti-symmetry component of z_(l−1,k), z_(l,k).

In systems like the ultrawide band (UWB) multiband OFDM frequencyhopping is allowed as a form of spectrum spreading, which opens up thepossibility of using diversity techniques in synchronization algorithms.

Eqs. (40), (41), (43) can be invoked to justify a diversity method thatcombines the observations x_(l,k) among all frequency bands used inspectrum hopping during γ consecutive OFDM symbols. In an OFDM systemphase variation on subcarrier k must be monitor according to (44) and(45), over the duration of γ consecutive OFDM symbols, γ>1. Let

$\begin{matrix}{X_{l,{k;\gamma}}{\sum\limits_{i = 1}^{\gamma}\; x_{l,{k|i}}}} & (56)\end{matrix}$where x_(l,k|i) is the sufficient statistic obtained while hopping inthe i-th hopped band, i=1, . . . , γ. With an obvious notation, what isbeing combined are the magnitudes |R_(l−1,k|i)|, per eq. (49). Combiningthese observations does not change the dimensionality of a sufficientstatistic derived from a κ-parameter exponential family. The newstatistic is sufficient and complete for Δφ_(k,(γ)) of (52), and

$\begin{matrix}{{\chi_{l,{k;\gamma}}} = {{{\cos\left( {\Delta\varphi}_{k,{(\gamma)}} \right)}{\sum\limits_{i = 1}^{\gamma}\;{R_{{l - 1},{k|i}}}^{2}}} + \mu_{l,{k;\gamma;R}}}} & (57) \\{{\chi_{l,{k;\gamma}}} = {{{\sin\left( {\Delta\varphi}_{k,{(\gamma)}} \right)}{\sum\limits_{i = 1}^{\gamma}\;{R_{{l - 1},{k|i}}}^{2}}} + \mu_{l,{k;\gamma;I}}}} & (58)\end{matrix}$or, using (49),

$\begin{matrix}{{{{\alpha^{2}\left( \phi_{k} \right)}\beta^{2}{\sigma_{a}^{2}\left( {\sum\limits_{i = 1}^{\gamma}\;{H_{k❘i}}^{2}} \right)}{\cos\left( {\Delta\varphi}_{k,{(\gamma)}} \right)}} + \mu_{l,{k;\gamma;R}}},} & (59) \\{{{\;\chi_{l,{k;\gamma}}} = {{{\alpha^{2}\left( \phi_{k} \right)}\beta^{2}{\sigma_{a}^{2}\left( {\sum\limits_{i = 1}^{\gamma}\;{H_{k❘i}}^{2}} \right)}{\sin\left( {\Delta\varphi}_{k,{(\gamma)}} \right)}} + \mu_{l,{k;\gamma;I}}}},} & (60)\end{matrix}$and the diversity combined statistic [

χ_(l,k;γ)

χ_(l,k;γ)]^(T) is a two-dimensional sufficient statistic forΔφ_(k,(γ))=γΔf″_(F)±γζ′k, see (52), which is also complete and minimalper reasoning detailed above.

Any estimators based on x_(l,k) can be used on the diversity combinedstatistic [

χ_(l,k;γ)

χ_(l,k;γ)]^(T), with the advantage given by diversity. Note that thediversity-combining-induced-summation of the noise terms in theobservations x_(l,k) do enable the applicability of the central limittheorem to infer that the sufficient statistic is Gaussian. Theapplicability of the CLT is mirrored in the use of an unbiased estimatorfor both ζ and Δf_(F)T_(u), which serves as the starting point in thederivation of their, respectively unique, MVUBEs; the applicability ofthe CLT is due to a similar addition of noise terms inherent in the formof the unbiased estimators that lead to the MVUBEs.

In effect, the diversity combined statistic [

χ_(l,k;γ)

χ_(l,k;γ)]^(T) lends itself very well to computing MVUBEs for both ζ andΔf_(F)T_(u). In some embodiments, the expectation operation averages outadditive noise.

In some embodiments, an MVUB estimator for CFO and SFO can be used.Consider the scalar function of a_(k),

$\begin{matrix}{{{b\left( a_{k} \right)}\mspace{14mu}\bullet\mspace{14mu}{\tan\left\lbrack {\frac{1}{2}\left( {{\Delta\varphi}_{k,{(1)}} - {\Delta\varphi}_{{- k},{(1)}}} \right)} \right\rbrack}} = {{\tan\left( {\zeta^{\prime}k} \right)}.}} & (61)\end{matrix}$

Consider the estimate

$\begin{matrix}{{\overset{\bullet}{b}\left( a_{k} \right)}\mspace{14mu}\bullet\mspace{14mu}{\frac{{x_{l,k}} - {x_{l,{- k}}}}{{x_{l,k}} - {x_{l,{- k}}}}.}} & (62)\end{matrix}$

By (32), averaging the ratio of differences between homologouscomponents of the sufficient statistics for subcarriers k, −k, can bedone by averaging the numerator and denominator separately; via (31),and whenever the magnitude of the Fourier transform is the same atsymmetric (around zero) frequencies, i.e. |z_(l,k)|=|z_(l,−k)|,

$\begin{matrix}\begin{matrix}{{E\left\{ {\overset{\bullet}{b}\left( a_{k} \right)} \right\}} = \frac{{{z_{l,k}}^{2}{\sin\left( {{\Delta\; f_{F}^{''}} + {\zeta^{\prime}k}} \right)}} - {{z_{l,{- k}}}^{2}\sin\mspace{14mu}\left( {{\Delta\; f_{F}^{''}} - {\zeta^{\prime}k}} \right)}}{{{z_{l,k}}^{2}{\cos\left( {{\Delta\; f_{F}^{''}} + {\zeta^{\prime}k}} \right)}} + {{z_{l,{- k}}}^{2}\cos\mspace{14mu}\left( {{\Delta\; f_{F}^{''}} - {\zeta^{\prime}k}} \right)}}} \\{= {\tan\left( {\zeta^{\prime}k} \right)}}\end{matrix} & (63)\end{matrix}$where the trigonometric identity

${\left( {{\sin\;\alpha} - {\sin\;\beta}} \right)/\left( {{\cos\;\alpha} + {\cos\;\beta}} \right)} = {\tan\left( {\frac{1}{2}\left( {\alpha - \beta} \right)} \right)}$was used. Clearly,

(a_(k)) is unbiased.

A symmetric magnitude of the Fourier transform is encountered in caseswhen, for example, the time domain signal corresponding to the frequencydomain signal z_(l,k) is real. A symmetric magnitude of the Fouriertransform may also be encountered when z_(l,k) represents the conjugatesymmetric component of a signal, or when z_(l,k) reflects a diversitycombining operation whereby the symmetry occurs because of asuperposition of several uncorrelated quantities. These examples are notintended to be exhaustive.

From the multiple angle formula

${{\tan\mspace{14mu}({nx})} = \frac{{\tan\left( {\left( {n - 1} \right)x} \right)}\tan\mspace{14mu} x}{1 - {{\tan\left( {\left( {n - 1} \right)x} \right)}\tan\mspace{14mu} x}}},$it is clear by induction that if tan((n−1)x)≈(n−1)tan x then tan(nx)≈ntan x—as long as n, x are not large enough that (n−1)tan² x becomescomparable with 1. The relative error in approximating tan(nx)≈n tan xis plotted in double logarithmic scale for 10⁻⁶≦ζ′≦10⁻⁴, with severalvalues for n as a parameter. It is clear that for a significantpractical range for ζ′, n, the approximation tan(nx)≈n tan x holds verywell.

Consider the new scalar function of a_(k)

$\begin{matrix}\begin{matrix}{{{B\left( a_{k} \right)}\mspace{14mu}\bullet\mspace{14mu}\frac{1}{k}{b\left( a_{k} \right)}} = {\frac{1}{k}{\tan\left\lbrack {\frac{1}{2}\left( {{\Delta\varphi}_{k,{(1)}} - {\Delta\varphi}_{{- k},{(1)}}} \right)} \right\rbrack}}} \\{= {{\tan\left( {\zeta^{\prime}k} \right)}/k}} \\{= {\tan\mspace{11mu}\zeta^{\prime}}}\end{matrix} & (64)\end{matrix}$1. and the estimate

$\begin{matrix}{{\overset{\bullet}{B}\left( a_{k} \right)} = {{{\overset{\bullet}{b}\left( a_{k} \right)}/k}\mspace{14mu}\bullet\mspace{14mu}\frac{1}{k}\frac{{x_{l,k}} - {x_{l,{- k}}}}{{x_{l,k}} + {x_{l,{- k}}}}}} & (65)\end{matrix}$

For which the mean is E{

(a_(k))}=tan ζ′, making

(a_(k)) an unbiased estimator for tan ζ′.

The set of pilot indices C provides a set of |C|/2 unbiased estimatesfor tan ζ′, based on the complete and minimal sufficient statisticsx_(l,k), k∈C; |C| denotes the cardinality of C. By the Rao-Blackwelltheorem, averaging the |C|/2 unbiased estimates for tan ζ′ leads to aminimum variance unbiased estimator (MVUBE) for tan ζ′, which is uniquebecause the sufficient statistic is complete. It is unique because thereis one and only one function of a complete (thus minimal) sufficientstatistic that is unbiased. Because it is unique, the MVUBE does achievethe Cramer-Rao bound. This means that all other estimators of tan ζ′ (orζ′, e.g. eq. (14) of Speth Part II,) are not optimal, in the sense thatthey are not uniformly better than all other estimators (i.e., they arenot MVUBEs).

The MVUBE for tan ζ′ is

$\begin{matrix}{{E\left\{ {{{\overset{\bullet}{B}\left( a_{k} \right)}❘x_{l,k}},{k \in C}} \right\}\mspace{14mu}\bullet\mspace{14mu}\tan\limits^{\bullet}\mspace{14mu}\zeta^{\prime}} = {\frac{2}{C}{\sum\limits_{k \in C_{+}}^{\;}\;{\frac{1}{k}\frac{{x_{l,k}} - {x_{l,{- k}}}}{{x_{l,k}} + {x_{l,{- k}}}}}}}} & (66)\end{matrix}$

Note that it is the estimator for tan ζ′ that is unbiased, rather thanthe estimator for ζ′ itself; in general, the ML estimator

□ arctan(E{

(a_(k))|x_(l,k),k∈C}) is biased (apply Jensen's inequality). However,whenever ζ′<0.5, and certainly in the range ζ′∈[10⁻⁵,10⁻⁴], which isrepresentative of most practical applications of OFDM,tan ζ′≈ζ′  (67)

And thereby the MVUBE for ζ′ is

$\begin{matrix}{{\overset{\bullet}{\zeta^{\prime}}}_{MVUB} = {\frac{2}{C}{\sum\limits_{k \in C_{+}}^{\;}\;{\frac{1}{k}\frac{{x_{l,k}} - {x_{l,{- k}}}}{{x_{l,k}} + {x_{l,{- k}}}}}}}} & (68)\end{matrix}$

Naturally, the MVUBE for ζ is

$\begin{matrix}{{\overset{\bullet}{\zeta}}_{MVUB} = {\frac{1}{{\pi\left( {1 + {N_{g}/N}} \right)}{C}}{\sum\limits_{k \in C_{+}}^{\;}\;{\frac{1}{k}\frac{{x_{l,k}} - {x_{l,{- k}}}}{{x_{l,k}} + {x_{l,{- k}}}}}}}} & (69)\end{matrix}$

One can contrast the MVUBE for ζ given in (69) with the estimator givenin eq. (14) of Speth Part II. Because there is one and only one functionof a complete (thus minimal) sufficient statistic that is unbiased, allestimators different from (69) are not optimal; that is, they are notuniformly better than all other estimators, i.e., they are not MVUBEs.Note that this does not prevent the expectation of the unbiasedestimator for Δφ_(±k,(1)) of (51) (rather than ζ, or Δf) from being aMVUBE, per Rao-Blackwell's theorem. In fact, the above derivations showthat the Moose article did implement the average of an unbiasedestimator of θ from (1) (see the Appendix in the Moose article), therebyin effect working with a MVUBE; what was not obvious was to correctlyidentify a MVUBE for ζ′ and Δf″_(F).

Having provided a MVUBE for ζ′, a similar reasoning leads to a MVUBE forΔf″_(F). Consider the scalar function of a_(k)c(a _(k))□ tan[½(Δφ_(k,(1))+Δφ_(−k,(1)))]=tan(Δf″ _(F))  (70)and the estimate

$\begin{matrix}{{\overset{\bullet}{c}\left( a_{k} \right)}\mspace{14mu}\bullet\mspace{14mu}\frac{{x_{l,k}} + {x_{l,{- k}}}}{{x_{l,k}} + {x_{l,{- k}}}}} & (71)\end{matrix}$

As before, (32) shows that averaging the ratio of differences betweenhomologous components of the sufficient statistics for subcarriers k,−k, can be done by averaging the numerator and denominator separately;via (31), and because the magnitude of the Fourier transform is the sameat symmetric (around zero) frequencies, i.e. |z_(l,k)|=|z_(l,−k)|,

$\begin{matrix}\begin{matrix}{{E\left\{ {\overset{\bullet}{c}\left( a_{k} \right)} \right\}} = \frac{{{z_{l,k}}^{2}{\sin\left( {{\Delta\; f_{F}^{''}} + {\zeta^{\prime}k}} \right)}} + {{z_{l,{- k}}}^{2}\sin\mspace{14mu}\left( {{\Delta\; f_{F}^{''}} - {\zeta^{\prime}k}} \right)}}{{{z_{l,k}}^{2}{\cos\left( {{\Delta\; f_{F}^{''}} + {\zeta^{\prime}k}} \right)}} + {{z_{l,{- k}}}^{2}\cos\mspace{14mu}\left( {{\Delta\; f_{F}^{''}} - {\zeta^{\prime}k}} \right)}}} \\{= {\tan\left( {\Delta\; f_{F}^{''}} \right)}}\end{matrix} & (72)\end{matrix}$where the trigonometric identity

${\left( {{\sin\;\alpha} + {\sin\;\beta}} \right)/\left( {{\cos\;\alpha} + {\cos\;\beta}} \right)} = {\tan\left( {\frac{1}{2}\left( {\alpha + \beta} \right)} \right)}$was used. Clearly,

(a_(k)) is unbiased and based on a sufficient statistic. By theRao-Blackwell theorem, averaging the |C|/2 unbiased estimates fortan(Δf″_(F)) leads to a minimum variance unbiased estimator (MVUBE) fortan Δf″_(F), which is unique because the sufficient statistic iscomplete. Because it is unique, the MVUBE does achieve the Cramer-Raobound. The MVUBE for tan Δf″_(F) is

$\begin{matrix}{{E\left\{ {{{\overset{\bullet}{c}\left( a_{k} \right)}❘x_{l,k}},{k \in C}} \right\}\mspace{14mu}\bullet\mspace{14mu}\tan\limits^{\bullet}\;\Delta\; f_{F}^{''}} = {\frac{2}{C}{\sum\limits_{k \in C_{+}}^{\;}\;\frac{{x_{l,k}} + {x_{l,{- k}}}}{{x_{l,k}} + {x_{l,{- k}}}}}}} & (73)\end{matrix}$

As before, via tan Δf″_(F)≈Δf″_(F), the MVUBE for Δf″_(F) is

$\begin{matrix}{{{\overset{\bullet}{\Delta}}_{F}^{\prime}}^{MVUB} = {{\overset{\bullet}{\Delta}f_{F}T_{u}^{MVUB}} = {\frac{1}{{\pi\left( {1 + {N_{g}/N}} \right)}{C}}{\sum\limits_{k \in C_{+}}^{\;}\;\frac{{x_{l,k}} + {x_{l,{- k}}}}{{x_{l,k}} + {x_{l,{- k}}}}}}}} & (74)\end{matrix}$

Again, the MVUBE for Δf″_(F) given in (74) is clearly distinct from theestimator given in eq. (14) of Speth Part II. It is stressed again thatthe MVUBEs derived above for both ζ and Δf_(F)T_(u) are novel, and neednot rely on the channel estimator—which has inherent errors and couldirreversibly degrade the performance of a synchronization algorithm thatneeds channel estimates.

When correctly implemented, the expectation operation averages out theadditive noise, including the ICI term n_(Ω;l,k) (or frequency offsetnoise, as ICI is called in Speth Part I), which degrades the SNR. Inaddition, combining the pilot observables, when possible, can furtherreduce the SNR degradation.

The above steps do not exhaust the ways for implementing this method, ora related apparatus; for example one can add, remove, or modify steps.In one alternate embodiment a step might be added to retrieve aconjugate symmetric component, or a conjugate asymmetric component ofthe observations or their corresponding statistic, or metric. In anotherembodiment the estimates might be performed before combining therelevant statistics or metrics, as for example in diversity combining,and the estimates combined thereafter. Or, the estimators in (72), (63)might be modified to account for the case when |z_(l,k)|≠|z_(l,−k)|. Orone can modify the differential metric or use an approximation thereof.

Combining operation may, in some embodiments, average out the additivenoise terms, including the ICI term n_(Ω;l,k) (or frequency offsetnoise, as ICI is sometimes called, e.g. in Speth, Part I), whichdegrades the SNR. Therefore, combining the pilot observables can reducethe SNR degradation due to imperfect carrier frequency synchronization.

In the case when a narrow-band in-band interferer is active within theband of a tolerated (unlicensed) wideband or ultrawide band OFDM device,e.g. a UWB device, it might be necessary or desirable to detect such anarrow band device, referred to as a victim device, in order to protectit by avoiding operation in a band that overlaps the band of the narrowband victim device. The OFDM feature of the tolerated device allows formitigation techniques such as tone nulling, in addition to simplyavoiding the band of the narrow band victim device. One example is a UWBdevice that must detect a WiMax device, and perform avoidance ormitigation techniques after having detected such a narrow band device.Another example is military radar operating within the band of a UWBdevice. These examples are not exhaustive as practical embodiments ofthe ideas described below. While in the case of the WiMax device itscarrier frequency and operation band is known a priori (relative to theband of the UWB device), the method described below does not assume thistype of side information.

Desirably, detection of a narrow band interferer, especially when itsexact spectral position is not known a priori, must exploit thestructure present in the tolerated device, monitor its whole bandwidthfor energy, rely on modules and functions already present in thetolerated device, and account for any symmetries of the tolerated deviceor asymmetries of the victim device.

One embodiment that relies on signals and metrics associated with thesynchronization of the OFDM tolerated device attempts to detectasymmetries in the magnitude spectrum of the signal observed by thetolerated device, via DFT at the receiver and subcarriers (tones)signals, and discern whether the asymmetric pattern is due to aninterferer as opposed to frequency selectivity. Due to the fact that theDFT is simply a sampling of the continuous time spectrum, and the samplepositions are the discrete frequencies used by the DFT of the OFDMwideband device, the narrow band signal is seen as another sinusoid, ortwo sinusoids. This may be the case, if, for example, victim is alsoOFDM and its discrete frequency bins do not correspond exactly to thoseof the tolerated wideband OFDM device, superimposed on the discretefrequency domain sample of the tolerated wideband OFDM device. Theanalysis does not change significantly, as a person of ordinary skill inthe art can recognize.

In some embodiments, the venue is offered by a estimator where theassumption |z_(l,k)|=|z_(l,−k)| is purposely not made at one stage ofthe method and algorithm. Specifically, the estimator is computedwithout assuming that |z_(l,k)|=|z_(l,−k)|, or trying to retrieve aconjugate symmetric component; let x₀□arctan(|z_(l,k)|²/|z_(l,−k)|²)∈[0, π/2], i.e. |z_(l,k)|²/|z_(l,−k)|²□tan x₀=sin x₀/cos x₀. Clearly the case corresponds to x₀=π/4. In thiscase, it can be shown, after some trigonometric and algebraicmanipulations, an estimator verifies that:

$\begin{matrix}\begin{matrix}{{E\left\{ {\overset{\bullet}{b}\left( a_{k} \right)} \right\}} = \frac{{{z_{l,k}}^{2}{\sin\left( {{\Delta\; f_{F}^{''}} + {\zeta^{\prime}k}} \right)}} - {{z_{l,{- k}}}^{2}\sin\mspace{14mu}\left( {{\Delta\; f_{F}^{''}} - {\zeta^{\prime}k}} \right)}}{{{z_{l,k}}^{2}{\cos\left( {{\Delta\; f_{F}^{''}} + {\zeta^{\prime}k}} \right)}} + {{z_{l,{- k}}}^{2}\cos\mspace{14mu}\left( {{\Delta\; f_{F}^{''}} - {\zeta^{\prime}k}} \right)}}} \\{= \frac{{\sin\mspace{14mu}\left( {{\Delta\; f_{F}^{''}} + {\zeta^{\prime}k}} \right)} - {\left( {{z_{l,{- k}}}^{2}/{z_{l,k}}^{2}} \right)\mspace{14mu}\sin\mspace{14mu}\left( {{\Delta\; f_{F}^{''}} - {\zeta^{\prime}k}} \right)}}{{\cos\mspace{14mu}\left( {{\Delta\; f_{F}^{''}} + {\zeta^{\prime}k}} \right)} + {\left( {{z_{l,{- k}}}^{2}/{z_{l,k}}^{2}} \right)\mspace{14mu}\cos\mspace{14mu}\left( {{\Delta\; f_{F}^{''}} - {\zeta^{\prime}k}} \right)}}} \\{= \frac{{\sin\mspace{14mu}\left( {{\Delta\; f_{F}^{''}} + {\zeta^{\prime}k}} \right)} - {\left( {\sin\mspace{14mu}{x_{0}/\cos}\mspace{14mu} x_{0}} \right)\mspace{14mu}\sin\mspace{14mu}\left( {{\Delta\; f_{F}^{''}} - {\zeta^{\prime}k}} \right)}}{{\cos\mspace{14mu}\left( {{\Delta\; f_{F}^{''}} + {\zeta^{\prime}k}} \right)} + {\left( {\sin\mspace{14mu}{x_{0}/\cos}\mspace{14mu} x_{0}} \right)\mspace{14mu}\cos\mspace{14mu}\left( {{\Delta\; f_{F}^{''}} - {\zeta^{\prime}k}} \right)}}} \\{= \frac{{{- \tan}\mspace{14mu}\Delta\; f^{''}\tan\mspace{14mu}\left( {x_{0} - {\pi/4}} \right)} + {\tan\mspace{14mu}\zeta^{\prime}k}}{1 + {\tan\mspace{14mu}\Delta\; f^{''}\tan\mspace{14mu}\left( {x_{0} - {\pi/4}} \right)\tan\mspace{14mu}\zeta^{\prime}k}}} \\{\approx {{{- \tan}\mspace{14mu}\Delta\; f^{''}\tan\mspace{14mu}\left( {x_{0} - {\pi/4}} \right)} + {\tan\mspace{14mu}\zeta^{\prime}k}}}\end{matrix} & (75)\end{matrix}$where the approximation is justified because a product of incrementalfactors is much smaller than any of the factors and than 1. Note thatthis expression becomes tan ζ′k, as in (63), when x₀=π/4.

Likewise, when |z_(l,k)|=|z_(l,−k)| is not assumed, an estimator can beshown to verify, after some trigonometric and algebraic manipulations,

$\begin{matrix}\begin{matrix}{{E\left\{ {\overset{\bullet}{c}\left( a_{k} \right)} \right\}} = \frac{{{z_{l,k}}^{2}{\sin\left( {{\Delta\; f_{F}^{''}} + {\zeta^{\prime}k}} \right)}} + {{z_{l,{- k}}}^{2}{\sin\left( {{\Delta\; f_{F}^{''}} - {\zeta^{\prime}k}} \right)}}}{{{z_{l,k}}^{2}{\cos\left( {{\Delta\; f_{F}^{''}} + {\zeta^{\prime}k}} \right)}} + {{z_{l,{- k}}}^{2}{\cos\left( {{\Delta\; f_{F}^{''}} - {\zeta^{\prime}k}} \right)}}}} \\{= \frac{{\sin\left( {{\Delta\; f_{F}^{''}} + {\zeta^{\prime}k}} \right)} + {\left( {{z_{l,{- k}}}^{2}/{z_{l,k}}^{2}} \right){\sin\left( {{\Delta\; f_{F}^{''}} - {\zeta^{\prime}k}} \right)}}}{{\cos\left( {{\Delta\; f_{F}^{''}} + {\zeta^{\prime}k}} \right)} + {\left( {{z_{l,{- k}}}^{2}/{z_{l,k}}^{2}} \right){\cos\left( {{\Delta\; f_{F}^{''}} - {\zeta^{\prime}k}} \right)}}}} \\{= \frac{{\sin\left( {{\Delta\; f_{F}^{''}} + {\zeta^{\prime}k}} \right)} + {\left( {\sin\;{x_{0}/\cos}\; x_{0}} \right){\sin\left( {{\Delta\; f_{F}^{''}} - {\zeta^{\prime}k}} \right)}}}{{\cos\left( {{\Delta\; f_{F}^{''}} + {\zeta^{\prime}k}} \right)} + {\left( {\sin\;{x_{0}/\cos}\; x_{0}} \right){\cos\left( {{\Delta\; f_{F}^{''}} - {\zeta^{\prime}k}} \right)}}}} \\{= \frac{{\tan\;\Delta\; f^{''}} - {\tan\;\zeta^{\prime}k\mspace{14mu}{\tan\left( {x_{0} - {\pi/4}} \right)}}}{1 + {\tan\;\Delta\; f^{''}\mspace{14mu}{\tan\left( {x_{0} - {\pi/4}} \right)}\tan\;\zeta^{\prime}k}}} \\{\approx {{\tan\;\Delta\; f^{''}} - {\tan\;\zeta^{\prime}k\;\tan\;\left( {x_{0} - {\pi/4}} \right)}}}\end{matrix} & (76)\end{matrix}$and the sum of (75), (76) is the mean of the summed estimators

$\begin{matrix}\begin{matrix}{{E\left\{ {{\overset{\bullet}{c}\left( a_{k} \right)} + {\overset{\bullet}{b}\left( a_{k} \right)}} \right\}} = {\left( {{\tan\;\Delta\; f^{''}} + {\tan\;\zeta^{\prime}k}} \right)\left( {1 - {\tan\left( {x_{0} - {\pi/4}} \right)}} \right)}} \\{= {\left( {{\tan\;\Delta\; f^{''}} + {k\;\tan\;\zeta^{\prime}}} \right)\;\left( {1 - {\tan\left( {x_{0} - {\pi/4}} \right)}} \right)}}\end{matrix} & (77)\end{matrix}$

Thereby, at each subcarrier k, the linear dependence tan Δf″+k tan ζ′determined as a result of synchronization based on a conjugate symmetricset of frequency observations is altered by the factor1−tan(x₀−π/4)—which, because tan (x₀−π/4)∈[−1, 1], takes values 0 when|z_(l,k)|□|z_(l,−k)|, 2 when |z_(l,k)|□|z_(l,−k)|, and 1 when|z_(l,k)|=|z_(l,−k)| (absent interferer with conjugate symmetricfrequency domain observations).

Note that x₀ contains information about the relative energy imbalance ontone k; the sum of the two estimators—possibly with furtherapproximations and simplifications, and possibly after removing thefactor tan Δf′+k tan ζ′ due only to the tolerated wideband device—can becompared with an appropriately selected threshold to discern whether theasymmetric pattern is due to an interferer as opposed to frequencyselectivity.

Further processing can be aimed at mapping 1−tan(x₀−π/4) to the relativestrength of the positive tone k vs. the negative tone −k.

It is this structure that can be exploited to obtain indications aboutthe presence of a narrowband interferer, and its strength relative tothe UWB device.

The frequency selective fading affecting the tones in the UWB, includinga model for correlation due to finite excess delay in the time domain,can be modeled with a pdf, and a threshold selected that guaranteesdesired probabilities of false alarm and detection.

FIG. 2, which comprises FIG. 2A and FIG. 2B is a flowchart illustratingan example system in accordance with the systems and methods describedherein. Referring now to FIG. 2A, in a step 200 a differential phasemetric is calculated. In some embodiments, the differential phase metricmay be calculated at pilot tone k between two consecutive OFDM symbolssent in the same frequency band; naturally, this translates intoobserving, in the frequency domain, at least two consecutive OFDMsymbols, or as many as the number of active spectral hop bands plus one.

In a step 202, the calculation is repeated for other pilot tones, or alltones that are to be used in estimating the carrier and sampling clockrelative frequency offsets. In some embodiments, for example, in adiversity based system, this may be done for all subcarriers that are tobe used in estimating the carrier and sampling clock relative frequencyoffsets. Additionally, in a step 204, if more than one diversitybranches are available, collect diversity observations on the same pilottones, during a number of uncorrelated diversity branches, e.g.frequency hops. Further, in systems with more than one diversitybranches diversity observations of the (sufficient) differentialsynchronization statistic are combined in a step 206.

In a step 208, an estimator is used. The unique minimum varianceunbiased estimator for the carrier and sampling clock relative frequencyoffsets is applied, as described above or with reasonablesimplifications. Some embodiments may work with a symmetricrepresentation of the frequency domain signal, such as a conjugatesymmetric component retrieved from the original observables. In someembodiments, the systems and methods described herein may process asymmetric representation of a frequency domain signal, such as a CScomponent retrieved from original observables. In various embodiments, avalid estimator for the carrier and sampling clock relative frequencyoffsets may be used.

In a step 210, other embodiments apply the unique minimum varianceunbiased estimator without resorting to a symmetric component such as aconjugate symmetric component retrieved from the original observables.

Continuing with FIG. 2B, in a step 212 estimates are combine. Theestimates may be combined for example, by summing them. Additionally, ina step 214, some embodiments may remove the effect of the widebandsignal from the combined estimates based on asymmetric observables. In astep 216 the systems and methods described herein predict a narrowbandinterferer. For example, some embodiments operate on the resultingmetric, in one embodiment by comparing it to a selected threshold, inorder to make the hypotheses that a narrow band interferer is present orabsent.

In a step 218, various embodiments process the metric due to a possiblenarrowband interferer in order to assess its relative strength withrespect to the tolerated wideband device, and to avoid or mitigate thenarrow band device operating within the band of the tolerated widebanddevice.

In a step 220, various embodiments correct the local oscillatorfrequency. For example, the systems and methods may correct the localoscillator(s) frequency, or form and apply a correction to thedemodulated samples (post DFT) based on the estimated synchronizationparameters. This correction may be done in parallel in some embodiments.

In some embodiments, the estimation may be repeated. Additionally, insome embodiments a detection can be repeated. For example, in a step 222the estimation and detection procedure may be repeated throughout theduration of the packet.

The above steps do not exhaust the ways for implementing this method, ora related apparatus; for example one can add, remove, or modify steps.In one alternate embodiment a step might be added to retrieve aconjugate symmetric component, or a conjugate asymmetric component ofthe observations or their corresponding statistic, or metric. In anotherembodiment the estimates might be performed before combining therelevant statistics or metrics, as for example in diversity combining,and the estimates combined thereafter. Or the estimators in (72), (63)might be modified to account for the case when |z_(l,k)|≠|z_(l,−k)|. Orone can modify the differential metric or use an approximation thereof.

An additional and important benefit is that the combining operation canaverage out the additive noise terms, including the ICI term n_(Ω;l,k)(or frequency offset noise, as ICI is sometimes called), which degradesthe SNR. Therefore, combining the pilot observables can reduce the SNRdegradation due to imperfect carrier frequency synchronization.

The systems and methods described herein may be implemented using acomputer. In one embodiment the computer may be a desktop, laptop, ornotebook computer. In another embodiment the computer may be amainframe, supercomputer or workstation. In yet another embodiment thecomputer may be a hand-held computing device such as a PDA, smart phone,cell phone 718, palmtop, etc. The computer may also represent computingcapabilities embedded within or otherwise available to a given device.

The computer may include one or more processors, which may bemicroprocessors, microcontrollers, or other control logic and memory,such as random access memory (“RAM”), read only memory (“ROM”) or otherstorage device for storing information and instructions for theprocessor. Other information storage mechanisms may also be connected tothe computer, such as a hard disk drive, a floppy disk drive, a magnetictape drive, an optical disk drive, a CD or DVD drive (R or RW), or otherremovable or fixed media drive, such as a program cartridge andcartridge interface, a removable memory (for example, a flash memory orother removable memory module) and memory slot, a PCMCIA slot and card,and other fixed or removable storage units and interfaces that allowsoftware and data to be transferred from the storage unit to thecomputer.

The computer may also include a communications interface that may beused to allow software and data to be transferred between the computerand external devices. Examples of the communications interface mayinclude a modem or softmodem, a network interface (such as an Ethernet,network interface card, or other interface), a communications port (suchas for example, a USB port, IR port, RS232 port or other port), or otherwired or wireless communications interface. Software and datatransferred via the communications interface are carried on signals,which may be electronic, electromagnetic, optical or other signalscapable of being received by a given communications interface. Thesignals may be provided to the communications interface using a wired orwireless medium. Some examples of a channel may include a phone line, acellular phone link, an RF link, an optical link, a network interface, alocal or wide area network, the internet, and other communicationschannels.

In this document, the terms “computer program medium” and “computerusable medium” are used to generally refer to media such as, forexample, the memory, storage unit, media, and signals on a channel.These and other various forms of computer usable media may be involvedin carrying one or more sequences of one or more instructions to theprocessor for execution. Such instructions, generally referred to as“computer program code” (which may be grouped in the form of computerprograms or other groupings), when executed, enable the computer toperform features or functions of the present invention as discussedherein.

While various embodiments of the present invention have been describedabove, it should be understood that they have been presented by way ofexample only, and not of limitation. Likewise, the various diagrams maydepict an example architectural or other configuration for theinvention, which is done to aid in understanding the features andfunctionality that may be included in the invention. The invention isnot restricted to the illustrated example architectures orconfigurations, but the desired features may be implemented using avariety of alternative architectures and configurations. Indeed, it willbe apparent to one of skill in the art how alternative functional,logical or physical partitioning and configurations may be implementedto implement the desired features of the present invention. Also, amultitude of different constituent module names other than thosedepicted herein may be applied to the various partitions. Additionally,with regard to flow diagrams, operational descriptions and methodclaims, the order in which the steps are presented herein shall notmandate that various embodiments be implemented to perform the recitedfunctionality in the same order unless the context dictates otherwise.

Although the invention is described above in terms of various exemplaryembodiments and implementations, it should be understood that thevarious features, aspects and functionality described in one or more ofthe individual embodiments are not limited in their applicability to theparticular embodiment with which they are described, but instead may beapplied, alone or in various combinations, to one or more of the otherembodiments of the invention, whether or not such embodiments aredescribed and whether or not such features are presented as being a partof a described embodiment. Thus the breadth and scope of the presentinvention should not be limited by any of the above-described exemplaryembodiments.

Terms and phrases used in this document, and variations thereof, unlessotherwise expressly stated, should be construed as open ended as opposedto limiting. As examples of the foregoing: the term “including” shouldbe read as meaning “including, without limitation” or the like; the term“example” is used to provide exemplary instances of the item indiscussion, not an exhaustive or limiting list thereof; the terms “a” or“an” should be read as meaning “at least one,” “one or more,” or thelike; and adjectives such as “conventional,” “traditional,” “normal,”“standard,” “known” and terms of similar meaning should not be construedas limiting the item described to a given time period or to an itemavailable as of a given time, but instead should be read to encompassconventional, traditional, normal, or standard technologies that may beavailable or known now or at any time in the future. Likewise, wherethis document refers to technologies that would be apparent or known toone of ordinary skill in the art, such technologies encompass thoseapparent or known to the skilled artisan now or at any time in thefuture.

A group of items linked with the conjunction “and” should not be read asrequiring that each and every one of those items be present in thegrouping, but rather should be read as “and/or” unless expressly statedotherwise. Similarly, a group of items linked with the conjunctionshould not be read as requiring mutual exclusivity among that group, butrather should also be read as “and/or” unless expressly statedotherwise. Furthermore, although items, elements or components of theinvention may be described or claimed in the singular, the plural iscontemplated to be within the scope thereof unless limitation to thesingular is explicitly stated.

The presence of broadening words and phrases such as “one or more,” “atleast,” “but not limited to” or other like phrases in some instancesshall not be read to mean that the narrower case is intended or requiredin instances where such broadening phrases may be absent. The use of theterm “module” does not imply that the components or functionalitydescribed or claimed as part of the module are all configured in acommon package. Indeed, any or all of the various components of amodule, whether control logic or other components, may be combined in asingle package or separately maintained and may further be distributedacross multiple locations.

Additionally, the various embodiments set forth herein are described interms of exemplary block diagrams, flow charts and other illustrations.As will become apparent to one of ordinary skill in the art afterreading this document, the illustrated embodiments and their variousalternatives can be implemented without confinement to the illustratedexamples. For example, block diagrams and their accompanying descriptionshould not be construed as mandating a particular architecture orconfiguration.

As used herein, the term module might describe a given unit offunctionality that can be performed in accordance with one or moreembodiments of the present invention. As used herein, a module might beimplemented utilizing any form of hardware, software, or a combinationthereof. For example, one or more processors, controllers, ASICs, PLAs,logical components or other mechanisms might be implemented to make up amodule. In implementation, the various modules described herein might beimplemented as discrete modules or the functions and features describedcan be shared in part or in total among one or more modules. In otherwords, as would be apparent to one of ordinary skill in the art afterreading this description, the various features and functionalitydescribed herein may be implemented in any given application and can beimplemented in one or more separate or shared modules in variouscombinations and permutations. Even though various features or elementsof functionality may be individually described or claimed as separatemodules, one of ordinary skill in the art will understand that thesefeatures and functionality can be shared among one or more commonsoftware and hardware elements, and such description shall not requireor imply that separate hardware or software components are used toimplement such features or functionality.

1. A method in an OFDM communications system implementing frequencyhopping, comprising: receiving a first pair of OFDM symbols on a firstfrequency band; receiving a second pair of OFDM symbols on a secondfrequency band; obtaining a first pair of frequency domainrepresentations of the first pair of OFDM symbols at a pair of pilottones; obtaining a second pair of frequency domain representations ofthe second pair of OFDM symbols at the pair of pilot tones; forming afirst frequency domain statistic for the first frequency band bymultiplying one of the first pair of frequency domain representationswith the complex conjugate of the other of the first pair of frequencydomain representations; forming a second frequency domain statistic forthe second frequency band by multiplying on of the second pair offrequency domain representations with the complex conjugate of the otherof the second pair of frequency domain representations; and combiningthe first frequency domain statistic with the second frequency domainstatistic to form a combined frequency domain statistic.
 2. The methodof claim 1, wherein the pair of pilot tones are pilot tones distributedsymmetrically about
 0. 3. The method of claim 1, wherein the combinedfrequency domain statistic is [

χ_(l,k;γ)

χ_(l,k;γ)]^(T), where${\chi_{l,{k;\gamma}}\bullet{\sum\limits_{i = 1}^{\gamma}\; x_{l,{k❘i}}}},$where the set {x_(l,k|i)}_(i=1) ^(γ) is a set of frequency domainstatistics on the pilot tone pair ±k, and includes the first frequencydomain statistic and the second frequency domain statistic, l is asymbol time index, and i=1, . . . , γ is a sequence of frequency bandscorresponding to the frequency hopping implemented by the OFDMcommunications system.
 4. The method of claim 3, further comprisingcomputing a minimum variance unbiased estimator including the term$\frac{{\chi_{l,{k;\gamma}}} - {\chi_{l,{k;\gamma}}}}{{\chi_{l,{k;\gamma}}} + {\chi_{l,{k;\gamma}}}}\mspace{14mu}{or}\mspace{14mu}{\frac{{\chi_{l,{k;\gamma}}} + {\chi_{l,{k;\gamma}}}}{{\chi_{l,{k;\gamma}}} + {\chi_{l,{k;\gamma}}}}.}$5. The method of claim 4, wherein the minimum variance unbiasedestimator comprises arctan$\left( \frac{{\chi_{l,{k;\gamma}}} - {\chi_{l,{k;\gamma}}}}{{\chi_{l,{k;\gamma}}} + {\chi_{l,{k;\gamma}}}} \right)$or arctan$\left( \frac{{\chi_{l,{k;\gamma}}} + {\chi_{l,{k;\gamma}}}}{{\chi_{l,{k;\gamma}}} + {\chi_{l,{k;\gamma}}}} \right).$6. The method of claim 1, further comprising forming combined frequencydomain statistics for the first and second pairs of OFDM symbols at aplurality of tone pairs.
 7. The method of claim 1, further comprisingusing the combined frequency domain statistic to generate a minimumvariance unbiased estimator computation for carrier frequency offset orsampling frequency offset at the tone pair for the first and secondpairs of OFDM symbols.
 8. The method of claim 7, further comprisingforming a plurality of minimum variance unbiased estimator computationsfor carrier frequency offset or sampling frequency offset at a pluralityof tone pairs for the first and second pairs of OFDM symbols.
 9. Themethod of claim 8, further comprising averaging the plurality of minimumvariance unbiased estimator computations.
 10. A system for use in anOFDM communications system implementing frequency hopping, comprising: atransceiver; a processor, coupled to the transceiver; and a memory,coupled to the processor and configured to store instructions causingthe processor to perform the steps of: receiving a first pair of OFDMsymbols on a first frequency band; receiving a second pair of OFDMsymbols on a second frequency band; obtaining a first pair of frequencydomain representations of the first pair of OFDM symbols at a pair ofpilot tones; obtaining a second pair of frequency domain representationsof the second pair of OFDM symbols at the pair of pilot tones; forming afirst frequency domain statistic for the first frequency band bymultiplying one of the first pair of frequency domain representationswith the complex conjugate of the other of the first pair of frequencydomain representations; forming a second frequency domain statistic forthe second frequency band by multiplying on of the second pair offrequency domain representations with the complex conjugate of the otherof the second pair of frequency domain representations; and combiningthe first frequency domain statistic with the second frequency domainstatistic to form a combined frequency domain statistic.
 11. The systemof claim 10, wherein the pair of pilot tones are pilot tones distributedsymmetrically about
 0. 12. The system of claim 10, wherein the combinedfrequency domain statistic is [

χ_(l,k;γ)

χ_(l,k;γ)]^(T), where${\chi_{l,{k;\gamma}}\bullet{\sum\limits_{i = 1}^{\gamma}\; x_{l,{k❘i}}}},$where the set {x_(l,k|i)}_(i=1) ^(γ) is a set of frequency domainstatistics on the pilot tone pair ±k, and includes the first frequencydomain statistic and the second frequency domain statistic, l is asymbol time index, and i=1, . . . , γ is a sequence of frequency bandscorresponding to the frequency hopping implemented by the OFDMcommunications system.
 13. The system of claim 12, wherein the stepsfurther comprise computing a minimum variance unbiased estimatorincluding the term$\frac{{\;\chi_{l,{k;\gamma}}} - {\chi_{l,{k;\gamma}}}}{{\chi_{l,{k;\gamma}}} + {\chi_{l,{k;\gamma}}}}$or$\frac{{\chi_{l,{k;\gamma}}} + {\chi_{l,{k;\gamma}}}}{{\chi_{l,{k;\gamma}}} + {\chi_{l,{k;\gamma}}}}.$14. The system of claim 13, wherein the minimum variance unbiasedestimator comprises arctan$\left( \frac{{\chi_{l,{k;\gamma}}} - {\chi_{l,{k;\gamma}}}}{{\chi_{l,{k;\gamma}}} + {\chi_{l,{k;\gamma}}}} \right)$or arctan$\left( \frac{{\chi_{l,{k;\gamma}}} + {\chi_{l,{k;\gamma}}}}{{\chi_{l,{k;\gamma}}} + {\chi_{l,{k;\gamma}}}} \right).$15. The system of claim 10, wherein the steps further comprise formingcombined frequency domain statistics for the first and second pairs ofOFDM symbols at a plurality of tone pairs.
 16. The system of claim 10,wherein the steps further comprise using the combined frequency domainstatistic to generate a minimum variance unbiased estimator computationfor carrier frequency offset or sampling frequency offset at the tonepair for the first and second pairs of OFDM symbols.
 17. The system ofclaim 16, wherein the steps further comprise forming a plurality ofminimum variance unbiased estimator computations for carrier frequencyoffset or sampling frequency offset at a plurality of tone pairs for thefirst and second pairs of OFDM symbols.
 18. The system of claim 17,wherein the steps further comprise averaging the plurality of minimumvariance unbiased estimator computations.